The Riemann zeta function
The Riemann zeta function ζ(s) is a function of a complex variable s = a + bi

Gamma


Dirichlet Eta Function [ Doc File ] Euler Product [ Doc File ] Gamma Function [ Doc File ] Infinite Product [ Doc File ] Reflection Formula [ Doc File ]
ETA Function Euler Product Gamma Function Infinite Product Reflection Formula
ETA Function Euler Product Gamma Function Infinite Product Reflection Formula
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Zeta Function Table (Simple Mode)

ζ(-20) = 1+20 + 2+20 + 3+20 + ... + n+20 = (0) + (1/21)*n21+(1/2)*n20+(1/12)(20)*n19+(0)n18-(1/120)(1140)*n17+(0)n16+(1/252)(15504)*n15+(0)n14-(1/240)(77520)*n13+(0)n12+(1/132)(167960)*n11+(0)n10-(691/32760)(167960)*n9 +(0)n8 +(1/12)(77520)*n7 +(0)n6 -(3617/8160)(15504)*n5 +(0)n4 +(43867/14364)(1140)*n3 +(0)n2 -(174611/6600)(20)*n1 +(0)n0
ζ(-19) = 1+19 + 2+19 + 3+19 + ... + n+19 = (174611/6600) + (1/20)*n20+(1/2)*n19+(1/12)(19)*n18+(0)n17-(1/120)( 969)*n16+(0)n15+(1/252)(11628)*n14+(0)n13-(1/240)(50388)*n12+(0)n11+(1/132)( 92378)*n10+(0)n9 -(691/32760)( 75582)*n8 +(0)n7 +(1/12)(27132)*n6 +(0)n5 -(3617/8160)( 3876)*n4 +(0)n3 +(43867/14364)( 171)*n2 +(0)n1 -(174611/6600)( 1)*n0
ζ(-18) = 1+18 + 2+18 + 3+18 + ... + n+18 = (0) + (1/19)*n19+(1/2)*n18+(1/12)(18)*n17+(0)n16-(1/120)( 816)*n15+(0)n14+(1/252)( 8568)*n13+(0)n12-(1/240)(31824)*n11+(0)n10+(1/132)( 48620)*n9 +(0)n8 -(691/32760)( 31824)*n7 +(0)n6 +(1/12)( 8568)*n5 +(0)n4 -(3617/8160)( 816)*n3 +(0)n2 +(43867/14364)( 18)*n1 +(0)n0
ζ(-17) = 1+17 + 2+17 + 3+17 + ... + n+17 = - (43867/14364) + (1/18)*n18+(1/2)*n17+(1/12)(17)*n16+(0)n15-(1/120)( 680)*n14+(0)n13+(1/252)( 6188)*n12+(0)n11-(1/240)(19448)*n10+(0)n9 +(1/132)( 24310)*n8 +(0)n7 -(691/32760)( 12376)*n6 +(0)n5 +(1/12)( 2380)*n4 +(0)n3 -(3617/8160)( 136)*n2 +(0)n1 +(43867/14364)( 1)*n0
ζ(-16) = 1+16 + 2+16 + 3+16 + ... + n+16 = (0) + (1/17)*n17+(1/2)*n16+(1/12)(16)*n15+(0)n14-(1/120)( 560)*n13+(0)n12+(1/252)( 4368)*n11+(0)n10-(1/240)(11440)*n9 +(0)n8 +(1/132)( 11440)*n7 +(0)n6 -(691/32760)( 4368)*n5 +(0)n4 +(1/12)( 560)*n3 +(0)n2 -(3617/8160)( 16)*n1 +(0)n0
ζ(-15) = 1+15 + 2+15 + 3+15 + ... + n+15 = (3617/8160) + (1/16)*n16+(1/2)*n15+(1/12)(15)*n14+(0)n13-(1/120)( 455)*n12+(0)n11+(1/252)( 3003)*n10+(0)n9 -(1/240)( 6435)*n8 +(0)n7 +(1/132)( 5005)*n6 +(0)n5 -(691/32760)( 1365)*n4 +(0)n3 +(1/12)( 105)*n2 +(0)n1 -(3617/8160)( 1)*n0
ζ(-14) = 1+14 + 2+14 + 3+14 + ... + n+14 = (0) + (1/15)*n15+(1/2)*n14+(1/12)(14)*n13+(0)n12-(1/120)( 364)*n11+(0)n10+(1/252)( 2002)*n9 +(0)n8 -(1/240)( 3432)*n7 +(0)n6 +(1/132)( 2002)*n5 +(0)n4 -(691/32760)( 364)*n3 +(0)n2 +(1/12)( 14)*n1 +(0)n0
ζ(-13) = 1+13 + 2+13 + 3+13 + ... + n+13 = - (1/12) + (1/14)*n14+(1/2)*n13+(1/12)(13)*n12+(0)n11-(1/120)( 286)*n10+(0)n9 +(1/252)( 1287)*n8 +(0)n7 -(1/240)( 1716)*n6 +(0)n5 +(1/132)( 715)*n4 +(0)n3 -(691/32760)( 78)*n2 +(0)n1 +(1/12)( 1)*n0
ζ(-12) = 1+12 + 2+12 + 3+12 + ... + n+12 = (0) + (1/13)*n13+(1/2)*n12+(1/12)(12)*n11+(0)n10-(1/120)( 220)*n9 +(0)n8 +(1/252)( 792)*n7 +(0)n6 -(1/240)( 792)*n5 +(0)n4 +(1/132)( 220)*n3 +(0)n2 -(691/32760)( 12)*n1 +(0)n0 = n(n+1)/2 * (2n+1)/3 * (105n10 +525n9 +525n8 -1050n7 -1190n6 +2310n5 +1420n4 -3285n3 -287n2 +2073n-691)/455
ζ(-11) = 1+11 + 2+11 + 3+11 + ... + n+11 = (691/32760) + (1/12)*n12+(1/2)*n11+(1/12)(11)*n10+(0)n9 -(1/120)( 165)*n8 +(0)n7 +(1/252)( 462)*n6 +(0)n5 -(1/240)( 330)*n4 +(0)n3 +(1/132)( 55)*n2 +(0)n1 -(691/32760)( 1)*n0 = n(n+1)/2 * n(n+1)/2 * (2n8 +8n7 +4n6 -16n5 -5n4 +26n3 -3n2 -20n+10)/6
ζ(-10) = 1+10 + 2+10 + 3+10 + ... + n+10 = (0) + (1/11)*n11+(1/2)*n10+(1/12)(10)*n9 +(0)n8 -(1/120)( 120)*n7 +(0)n6 +(1/252)( 252)*n5 +(0)n4 -(1/240)( 120)*n3 +(0)n2 +(1/132)( 10)*n1 +(0)n0 = n(n+1)/2 * (2n+1)/3 * (n2 +n-1) * (3n6 +9n5 +2n4 -11n3 +3n2 +10n-5)/11
ζ(-9) = 1+9 + 2+9 + 3+9 + ... + n+9 = - (1/132) + (1/10)*n10+(1/2)*n9 +(1/12)( 9)*n8 +(0)n7 -(1/120)( 84)*n6 +(0)n5 +(1/252)( 126)*n4 +(0)n3 -(1/240)( 36)*n2 +(0)n1 +(1/132)( 1)*n0 = n(n+1)/2 * n(n+1)/2 * (n2 +n-1) * (2n4 +4n3 -n2 -3n+3)/5
ζ(-8) = 1+8 + 2+8 + 3+8 + ... + n+8 = (0) + (1/9 )*n9 +(1/2)*n8 +(1/12)( 8)*n7 +(0)n6 -(1/120)( 56)*n5 +(0)n4 +(1/252)( 56)*n3 +(0)n2 -(1/240)( 8)*n1 +(0)n0 = n(n+1)/2 * (2n+1)/3 * (5n6 +15n5 +5n4 -15n3 -n2 +9n-3)/15
ζ(-7) = 1+7 + 2+7 + 3+7 + ... + n+7 = (1/240) + (1/8 )*n8 +(1/2)*n7 +(1/12)( 7)*n6 +(0)n5 -(1/120)( 35)*n4 +(0)n3 +(1/252)( 21)*n2 +(0)n1 -(1/240)( 1)*n0 = n(n+1)/2 * n(n+1)/2 * (3n4 + 6n3 -n2 -4n-2)/6
ζ(-6) = 1+6 + 2+6 + 3+6 + ... + n+6 = (0) + (1/7 )*n7 +(1/2)*n6 +(1/12)( 6)*n5 +(0)n4 -(1/120)( 20)*n3 +(0)n2 +(1/252)( 6)*n1 +(0)n0 = n(n+1)/2 * (2n+1)/3 * (3n4 + 6n3 -3n+1)/7
ζ(-5) = 1+5 + 2+5 + 3+5 + ... + n+5 = - (1/252) + (1/6 )*n6 +(1/2)*n5 +(1/12)( 5)*n4 +(0)n3 -(1/120)( 10)*n2 +(0)n1 +(1/252)( 1)*n0 = n(n+1)/2 * n(n+1)/2 * (2n2 + 2n-1)/3
ζ(-4) = 1+4 + 2+4 + 3+4 + ... + n+4 = (0) + (1/5 )*n5 +(1/2)*n4 +(1/12)( 4)*n3 +(0)n2 -(1/120)( 4)*n1 +(0)n0 = n(n+1)/2 * (2n+1)/3 * (3n2 + 3n-1)/5
ζ(-3) = 1+3 + 2+3 + 3+3 + ... + n+3 = (1/120) + (1/4 )*n4 +(1/2)*n3 +(1/12)( 3)*n2 +(0)n1 -(1/120)( 1)*n0 = n(n+1)/2 * n(n+1)/2
ζ(-2) = 1+2 + 2+2 + 3+2 + ... + n+2 = (0) + (1/3 )*n3 +(1/2)*n2 +(1/12)( 2)*n1 +(0)n0 = n(n+1)/2 * (2n+1)/3
ζ(-1) = 1+1 + 2+1 + 3+1 + ... + n+1 = - (1/12) + (1/2 )*n2 +(1/2)*n1 +(1/12)( 1)*n0 = n(n+1)/2
ζ( 0) = 10 + 20 + 30 + ... + n0 = - (1/2) + (1/1 )*n1 +(1/2)*n0 = n


Faulhaber's Formula [ Doc File ] My Correction For
Bernoulli Numbers [ Doc File ]
Faulhabers_Formula Bernoulli_Numbers
Faulhabers_Formula Bernoulli_Numbers
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Zeta Function Table (Advanced Mode)

Here is my Advanced Mode Table:

ζ(-20) = 1+20 + 2+20 + 3+20 + ... + n+20 = - (0) + (-1)(-(-1)!)((20)!/(20-(-1))!/(-1)!)*n20-(-1)+(1/2)*n20-0+(1/12)(20!/(20-1)!/1!)*n20-1+(0)n20-2+(-1)(1/120)(20!/(20-3)!/3!)*n20-3+(0)n20-4+(1/252)(20!/(20-5)!/5!)*n20-5+(0)n20-6+(-1)(1/240)(20!/(20-7)!/7!)*n20-7+(0)n20-8+(1/132)(20!/(20-9)!/9!)*n20-9+(0)n20-10+(-1)(691/32760)(20!/(20-11)!/11!)*n20-11+(0)n20-12+(1/12)(20!/(20-13)!/13!)*n20-13+(0)n20-14+(-1)(3617/8160)(20!/(20-15)!/15!)*n20-15+(0)n20-16+(43867/14364)(20!/(20-17)!/17!)*n20-17+(0)n20-18+(-1)(174611/6600)(20!/(20-19)!/19!)*n20-19+(0)n20-20
ζ(-19) = 1+19 + 2+19 + 3+19 + ... + n+19 = - (-1)(174611/6600) + (-1)(-(-1)!)((19)!/(19-(-1))!/(-1)!)*n19-(-1)+(1/2)*n19-0+(1/12)(19!/(19-1)!/1!)*n19-1+(0)n19-2+(-1)(1/120)(19!/(19-3)!/3!)*n19-3+(0)n19-4+(1/252)(19!/(19-5)!/5!)*n19-5+(0)n19-6+(-1)(1/240)(19!/(19-7)!/7!)*n19-7+(0)n19-8+(1/132)(19!/(19-9)!/9!)*n19-9+(0)n19-10+(-1)(691/32760)(19!/(19-11)!/11!)*n19-11+(0)n19-12+(1/12)(19!/(19-13)!/13!)*n19-13+(0)n19-14+(-1)(3617/8160)(19!/(19-15)!/15!)*n19-15+(0)n19-16+(43867/14364)(19!/(19-17)!/17!)*n19-17+(0)n19-18+(-1)(174611/6600)(19!/(19-19)!/19!)*n19-19
ζ(-18) = 1+18 + 2+18 + 3+18 + ... + n+18 = - (0) + (-1)(-(-1)!)((18)!/(18-(-1))!/(-1)!)*n18-(-1)+(1/2)*n18-0+(1/12)(18!/(18-1)!/1!)*n18-1+(0)n18-2+(-1)(1/120)(18!/(18-3)!/3!)*n18-3+(0)n18-4+(1/252)(18!/(18-5)!/5!)*n18-5+(0)n18-6+(-1)(1/240)(18!/(18-7)!/7!)*n18-7+(0)n18-8+(1/132)(18!/(18-9)!/9!)*n18-9+(0)n18-10+(-1)(691/32760)(18!/(18-11)!/11!)*n18-11+(0)n18-12+(1/12)(18!/(18-13)!/13!)*n18-13+(0)n18-14+(-1)(3617/8160)(18!/(18-15)!/15!)*n18-15+(0)n18-16+(43867/14364)(18!/(18-17)!/17!)*n18-17+(0)n18-18
ζ(-17) = 1+17 + 2+17 + 3+17 + ... + n+17 = - (43867/14364) + (-1)(-(-1)!)((17)!/(17-(-1))!/(-1)!)*n17-(-1)+(1/2)*n17-0+(1/12)(17!/(17-1)!/1!)*n17-1+(0)n17-2+(-1)(1/120)(17!/(17-3)!/3!)*n17-3+(0)n17-4+(1/252)(17!/(17-5)!/5!)*n17-5+(0)n17-6+(-1)(1/240)(17!/(17-7)!/7!)*n17-7+(0)n17-8+(1/132)(17!/(17-9)!/9!)*n17-9+(0)n17-10+(-1)(691/32760)(17!/(17-11)!/11!)*n17-11+(0)n17-12+(1/12)(17!/(17-13)!/13!)*n17-13+(0)n17-14+(-1)(3617/8160)(17!/(17-15)!/15!)*n17-15+(0)n17-16+(43867/14364)(17!/(17-17)!/17!)*n17-17
ζ(-16) = 1+16 + 2+16 + 3+16 + ... + n+16 = - (0) + (-1)(-(-1)!)((16)!/(16-(-1))!/(-1)!)*n16-(-1)+(1/2)*n16-0+(1/12)(16!/(16-1)!/1!)*n16-1+(0)n16-2+(-1)(1/120)(16!/(16-3)!/3!)*n16-3+(0)n16-4+(1/252)(16!/(16-5)!/5!)*n16-5+(0)n16-6+(-1)(1/240)(16!/(16-7)!/7!)*n16-7+(0)n16-8+(1/132)(16!/(16-9)!/9!)*n16-9+(0)n16-10+(-1)(691/32760)(16!/(16-11)!/11!)*n16-11+(0)n16-12+(1/12)(16!/(16-13)!/13!)*n16-13+(0)n16-14+(-1)(3617/8160)(16!/(16-15)!/15!)*n16-15+(0)n16-16
ζ(-15) = 1+15 + 2+15 + 3+15 + ... + n+15 = - (-1)(3617/8160) + (-1)(-(-1)!)((15)!/(15-(-1))!/(-1)!)*n15-(-1)+(1/2)*n15-0+(1/12)(15!/(15-1)!/1!)*n15-1+(0)n15-2+(-1)(1/120)(15!/(15-3)!/3!)*n15-3+(0)n15-4+(1/252)(15!/(15-5)!/5!)*n15-5+(0)n15-6+(-1)(1/240)(15!/(15-7)!/7!)*n15-7+(0)n15-8+(1/132)(15!/(15-9)!/9!)*n15-9+(0)n15-10+(-1)(691/32760)(15!/(15-11)!/11!)*n15-11+(0)n15-12+(1/12)(15!/(15-13)!/13!)*n15-13+(0)n15-14+(-1)(3617/8160)(15!/(15-15)!/15!)*n15-15
ζ(-14) = 1+14 + 2+14 + 3+14 + ... + n+14 = - (0) + (-1)(-(-1)!)((14)!/(14-(-1))!/(-1)!)*n14-(-1)+(1/2)*n14-0+(1/12)(14!/(14-1)!/1!)*n14-1+(0)n14-2+(-1)(1/120)(14!/(14-3)!/3!)*n14-3+(0)n14-4+(1/252)(14!/(14-5)!/5!)*n14-5+(0)n14-6+(-1)(1/240)(14!/(14-7)!/7!)*n14-7+(0)n14-8+(1/132)(14!/(14-9)!/9!)*n14-9+(0)n14-10+(-1)(691/32760)(14!/(14-11)!/11!)*n14-11+(0)n14-12+(1/12)(14!/(14-13)!/13!)*n14-13+(0)n14-14
ζ(-13) = 1+13 + 2+13 + 3+13 + ... + n+13 = - (1/12) + (-1)(-(-1)!)((13)!/(13-(-1))!/(-1)!)*n13-(-1)+(1/2)*n13-0+(1/12)(13!/(13-1)!/1!)*n13-1+(0)n13-2+(-1)(1/120)(13!/(13-3)!/3!)*n13-3+(0)n13-4+(1/252)(13!/(13-5)!/5!)*n13-5+(0)n13-6+(-1)(1/240)(13!/(13-7)!/7!)*n13-7+(0)n13-8+(1/132)(13!/(13-9)!/9!)*n13-9+(0)n13-10+(-1)(691/32760)(13!/(13-11)!/11!)*n13-11+(0)n13-12+(1/12)(13!/(13-13)!/13!)*n13-13
ζ(-12) = 1+12 + 2+12 + 3+12 + ... + n+12 = - (0) + (-1)(-(-1)!)((12)!/(12-(-1))!/(-1)!)*n12-(-1)+(1/2)*n12-0+(1/12)(12!/(12-1)!/1!)*n12-1+(0)n12-2+(-1)(1/120)(12!/(12-3)!/3!)*n12-3+(0)n12-4+(1/252)(12!/(12-5)!/5!)*n12-5+(0)n12-6+(-1)(1/240)(12!/(12-7)!/7!)*n12-7+(0)n12-8+(1/132)(12!/(12-9)!/9!)*n12-9+(0)n12-10+(-1)(691/32760)(12!/(12-11)!/11!)*n12-11+(0)n12-12 = n(n+1)/2 * (2n+1)/3 * (105n10 +525n9 +525n8 -1050n7 -1190n6 +2310n5 +1420n4 -3285n3 -287n2 +2073n-691)/455
ζ(-11) = 1+11 + 2+11 + 3+11 + ... + n+11 = - (-1)(691/32760) + (-1)(-(-1)!)((11)!/(11-(-1))!/(-1)!)*n11-(-1)+(1/2)*n11-0+(1/12)(11!/(11-1)!/1!)*n11-1+(0)n11-2+(-1)(1/120)(11!/(11-3)!/3!)*n11-3+(0)n11-4+(1/252)(11!/(11-5)!/5!)*n11-5+(0)n11-6+(-1)(1/240)(11!/(11-7)!/7!)*n11-7+(0)n11-8+(1/132)(11!/(11-9)!/9!)*n11-9+(0)n11-10+(-1)(691/32760)(11!/(11-11)!/11!)*n11-11 = n(n+1)/2 * n(n+1)/2 * (2n8 +8n7 +4n6 -16n5 -5n4 +26n3 -3n2 -20n+10)/6
ζ(-10) = 1+10 + 2+10 + 3+10 + ... + n+10 = - (0) + (-1)(-(-1)!)((10)!/(10-(-1))!/(-1)!)*n10-(-1)+(1/2)*n10-0+(1/12)(10!/(10-1)!/1!)*n10-1+(0)n10-2+(-1)(1/120)(10!/(10-3)!/3!)*n10-3+(0)n10-4+(1/252)(10!/(10-5)!/5!)*n10-5+(0)n10-6+(-1)(1/240)(10!/(10-7)!/7!)*n10-7+(0)n10-8+(1/132)(10!/(10-9)!/9!)*n10-9+(0)n10-10 = n(n+1)/2 * (2n+1)/3 * (n2 +n-1) * (3n6 +9n5 +2n4 -11n3 +3n2 +10n-5)/11
ζ(-9) = 1+9 + 2+9 + 3+9 + ... + n+9 = - (1/132) + (-1)(-(-1)!)(( 9)!/( 9-(-1))!/(-1)!)*n 9-(-1)+(1/2)*n 9-0+(1/12)( 9!/( 9-1)!/1!)*n 9-1+(0)n 9-2+(-1)(1/120)( 9!/( 9-3)!/3!)*n 9-3+(0)n 9-4+(1/252)( 9!/( 9-5)!/5!)*n 9-5+(0)n 9-6+(-1)(1/240)( 9!/( 9-7)!/7!)*n 9-7+(0)n 9-8+(1/132)( 9!/( 9-9)!/9!)*n 9-9 = n(n+1)/2 * n(n+1)/2 * (n2 +n-1) * (2n4 +4n3 -n2 -3n+3)/5
ζ(-8) = 1+8 + 2+8 + 3+8 + ... + n+8 = - (0) + (-1)(-(-1)!)(( 8)!/( 8-(-1))!/(-1)!)*n 8-(-1)+(1/2)*n 8-0+(1/12)( 8!/( 8-1)!/1!)*n 8-1+(0)n 8-2+(-1)(1/120)( 8!/( 8-3)!/3!)*n 8-3+(0)n 8-4+(1/252)( 8!/( 8-5)!/5!)*n 8-5+(0)n 8-6+(-1)(1/240)( 8!/( 8-7)!/7!)*n 8-7+(0)n 8-8 = n(n+1)/2 * (2n+1)/3 * (5n6 +15n5 +5n4 -15n3 -n2 +9n-3)/15
ζ(-7) = 1+7 + 2+7 + 3+7 + ... + n+7 = - (-1)(1/240) + (-1)(-(-1)!)(( 7)!/( 7-(-1))!/(-1)!)*n 7-(-1)+(1/2)*n 7-0+(1/12)( 7!/( 7-1)!/1!)*n 7-1+(0)n 7-2+(-1)(1/120)( 7!/( 7-3)!/3!)*n 7-3+(0)n 7-4+(1/252)( 7!/( 7-5)!/5!)*n 7-5+(0)n 7-6+(-1)(1/240)( 7!/( 7-7)!/7!)*n 7-7 = n(n+1)/2 * n(n+1)/2 * (3n4 + 6n3 -n2 -4n-2)/6
ζ(-6) = 1+6 + 2+6 + 3+6 + ... + n+6 = - (0) + (-1)(-(-1)!)(( 6)!/( 6-(-1))!/(-1)!)*n 6-(-1)+(1/2)*n 6-0+(1/12)( 6!/( 6-1)!/1!)*n 6-1+(0)n 6-2+(-1)(1/120)( 6!/( 6-3)!/3!)*n 6-3+(0)n 6-4+(1/252)( 6!/( 6-5)!/5!)*n 6-5+(0)n 6-6 = n(n+1)/2 * (2n+1)/3 * (3n4 + 6n3 -3n+1)/7
ζ(-5) = 1+5 + 2+5 + 3+5 + ... + n+5 = - (1/252) + (-1)(-(-1)!)(( 5)!/( 5-(-1))!/(-1)!)*n 5-(-1)+(1/2)*n 5-0+(1/12)( 5!/( 5-1)!/1!)*n 5-1+(0)n 5-2+(-1)(1/120)( 5!/( 5-3)!/3!)*n 5-3+(0)n 5-4+(1/252)( 5!/( 5-5)!/5!)*n 5-5 = n(n+1)/2 * n(n+1)/2 * (2n2 + 2n-1)/3
ζ(-4) = 1+4 + 2+4 + 3+4 + ... + n+4 = - (0) + (-1)(-(-1)!)(( 4)!/( 4-(-1))!/(-1)!)*n 4-(-1)+(1/2)*n 4-0+(1/12)( 4!/( 4-1)!/1!)*n 4-1+(0)n 4-2+(-1)(1/120)( 4!/( 4-3)!/3!)*n 4-3+(0)n 4-4 = n(n+1)/2 * (2n+1)/3 * (3n2 + 3n-1)/5
ζ(-3) = 1+3 + 2+3 + 3+3 + ... + n+3 = - (-1)(1/120) + (-1)(-(-1)!)(( 3)!/( 3-(-1))!/(-1)!)*n 3-(-1)+(1/2)*n 3-0+(1/12)( 3!/( 3-1)!/1!)*n 3-1+(0)n 3-2+(-1)(1/120)( 3!/( 3-3)!/3!)*n 3-3 = n(n+1)/2 * n(n+1)/2
ζ(-2) = 1+2 + 2+2 + 3+2 + ... + n+2 = - (0) + (-1)(-(-1)!)(( 2)!/( 2-(-1))!/(-1)!)*n 2-(-1)+(1/2)*n 2-0+(1/12)( 2!/( 2-1)!/1!)*n 2-1+(0)n 2-2 = n(n+1)/2 * (2n+1)/3
ζ(-1) = 1+1 + 2+1 + 3+1 + ... + n+1 = - (1/12) + (-1)(-(-1)!)(( 1)!/( 1-(-1))!/(-1)!)*n 1-(-1)+(1/2)*n 1-0+(1/12)( 1!/( 1-1)!/1!)*n 1-1 = n(n+1)/2
ζ( 0) = 1 0 + 2 0 + 3 0 + ... + n 0 = - (1/2) + (-1)(-(-1)!)(( 0)!/( 0-(-1))!/(-1)!)*n 0-(-1)+(1/2)*n 0-0 = n
ζ(+1) = 1-1 + 2-1 + 3-1 + ... + n-1 = - (-1)(-(-1 )!) + (-1)(-(-1)!)((-1)!/(-1-(-1))!/(-1)!)*n-1-(-1) + ln(n+1)-1/(2n+2)+0.57721566490153286060651209008240243104215933593992359880576723488486772677766467093694 ... A001620
ζ(+2) = 1-2 + 2-2 + 3-2 + ... = π+2 * 2+1 /( 1 )! * (1/12) = π2 / 6 = 1.64493406684822643647241516664602518921894990120679843773555822937000747040320087383362890061975870 ... A013661
ζ(+3) = 1-3 + 2-3 + 3-3 + ... = π+3 * 2+2 /( 2 )! * (0) + ??? = π3 / 25.7943501666186840185586365793965132900509523271312260706140213406494349134925061412251 ... A308637 = 1.20205690315959428539973816151144999076498629234049888179227155534183820578631309018645587360933525 ... A002117
ζ(+4) = 1-4 + 2-4 + 3-4 + ... = π+4 * 2+3 /( 3 )! * (1/120) = π4 / 90 = 1.08232323371113819151600369654116790277475095191872690768297621544412061618696884655690963594169991 ... A013662
ζ(+5) = 1-5 + 2-5 + 3-5 + ... = π+5 * 2+4 /( 4 )! * (0) + ??? = π5 / 295.121509929078814295416301676822594619632418745885100174880081881222512573492113833455 ... A309926 = 1.03692775514336992633136548645703416805708091950191281197419267790380358978628148456004310655713333 ... A013663
ζ(+6) = 1-6 + 2-6 + 3-6 + ... = π+6 * 2+5 /( 5 )! * (1/252) = π6 / 945 = 1.01734306198444913971451792979092052790181749003285356184240866400433218290195789788277397793853517 ... A013664
ζ(+7) = 1-7 + 2-7 + 3-7 + ... = π+7 * 2+6 /( 6 )! * (0) + ??? = π7 / 2995.28476444062987421457140194123586447237619811128862116034993083589922581051107464452 ... A309927 = 1.00834927738192282683979754984979675959986356056523870641728313657160147831735573534609696891385132 ... A013665
ζ(+8) = 1-8 + 2-8 + 3-8 + ... = π+8 * 2+7 /( 7 )! * (1/240) = π8 / 9450 = 1.00407735619794433937868523850865246525896079064985002032911020265258295257474881439528723037237197 ... A013666
ζ(+9) = 1-9 + 2-9 + 3-9 + ... = π+9 * 2+8 /( 8 )! * (0) + ??? = π9 / 29749.3509504167924732263575439992360954535708605981514652679131630981776684624977358377 ... A309928 = 1.00200839282608221441785276923241206048560585139488875654859661590978505339025839895039306912716958 ... A013667
ζ(+10) = 1-10 + 2-10 + 3-10 + ... = π+10 * 2+9 /( 9 )! * (1/132) = π10 / 93555 = 1.00099457512781808533714595890031901700601953156447751725778899463629146515191295439704196861038565 ... A013668
ζ(+11) = 1-11 + 2-11 + 3-11 + ... = π+11 * 2+10 /( 10)! * (0) + ??? = π11 / 294058.697516635663068056032177491189612189560972448164117512566969938747449053262053487 ... A309929 = 1.00049418860411946455870228252646993646860643575820861711914143610005405979821981470259184302356062 ... A013669
ζ(+12) = 1-12 + 2-12 + 3-12 + ... = π+12 * 2+11 /( 11)! * (691/32760) = π12 * 691 / 638512875 = 1.00024608655330804829863799804773967096041608845800340453304095213325201968194091304904280855190069 ... A013670
ζ(+13) = 1-13 + 2-13 + 3-13 + ... = π+13 * 2+12 /( 12)! * (0) + ??? = π13 / 2903320.99437496874471612902548598299518022850873348106519286211097791175125276089735094 ... No OEIS = 1.00012271334757848914675183652635739571427510589550984513670267162089672682984420981289271395326813 ... A013671
ζ(+14) = 1-14 + 2-14 + 3-14 + ... = π+14 * 2+13 /( 13)! * (1/12) = π14 * 2 / 18243225 = 1.00006124813505870482925854510513533374748169616915454948275520225286294102317742087665978297199846 ... A013672
ζ(+15) = 1-15 + 2-15 + 3-15 + ... = π+15 * 2+14 /( 14)! * (0) + ??? = π15 / 28657269.3940598590044202589379919803466424134329335109381917049703719697921088276545668 ... No OEIS = 1.00003058823630702049355172851064506258762794870685817750656993289333226715634227957307233434701754 ... A013673
ζ(+16) = 1-16 + 2-16 + 3-16 + ... = π+16 * 2+15 /( 15)! * (3617/8160) = π16 * 3617 / 325641566250 = 1.00001528225940865187173257148763672202323738899047153115310520358878708702795315178628560484632246 ... A013674
ζ(+17) = 1-17 + 2-17 + 3-17 + ... = π+17 * 2+16 /( 16)! * (0) + ??? = π17 / 282842403.463197426131307236264129094363182272952265735576995225855515620331164084358670 ... No OEIS = 1.00000763719763789976227360029356302921308824909026267909537984397293564329028245934208173863691667 ... A013675
ζ(+18) = 1-18 + 2-18 + 3-18 + ... = π+18 * 2+17 /( 17)! * (43867/14364) = π18 * 43867 / 38979295480125 = 1.00000381729326499983985646164462193973045469721895333114317442998763003954265004563800196866898964 ... A013676
ζ(+19) = 1-19 + 2-19 + 3-19 + ... = π+19 * 2+18 /( 18)! * (0) + ??? = π19 / 2791558622.71018270391989516441857455178217039199704213989473442616757883445034218379359 ... No OEIS = 1.00000190821271655393892565695779510135325857114483863023593304676182394970534130931266422711807630 ... A013677
ζ(+20) = 1-20 + 2-20 + 3-20 + ... = π+20 * 2+19 /( 19)! * (174611/6600) = π20 * 174611 / 1531329465290625 = 1.00000095396203387279611315203868344934594379418741059575005648985113751373114390025783609797638747 ... A013678

i did a correction for Bernoulli Numbers emphasis on the first starting -1 index which is -(-1)!
that result i got by expanding the coefficient 1/(m+1) to the right form (-1)(B-1)(m!/(m-(-1))!/(-1)!)
i am also geting the right result of -1/2 by using the expanding result -(-1)! in the functional equation

i agree that the next step could be a controversial step but it seems to work ... that why i put it here
this is not a proof but only a way to show you that it seem to work as well. i know (-1)! is undefined!
but also Imaginary number were undefined at some point and now you are dividing imaginary numbers as well

Important Note: -(-1)! is not the sum of the harmonic series its the analytic continuation value of ζ(1)
(which is a nice way to show you a pole point)

My_Functional_Equation

Zeta Function Table (Experimental Mode)

Here is my Experimental Mode Table:

this is how i see it! i hope you will enjoy and appreciate it. this was hard to make like the way it is now in it's final form
Please notice my correction for bernoulli numbers! my index for B start form -1 to n-1
Please notice that b-1 = -(-1)! which represents the pole at 1
also make sure you see the trick i did at ζ( 0) line how i set all the even places to zero
and i highlighted the (2-1) part π 0 *(2-1) /(-1 )! * (-(-1 )!) - (0) that implies from where the 1/2 "comes" from

ζ(-20) = 1+20 + 2+20 + 3+20 + ... + n+20 = π-20 * 2-21 /(-21)! * (±(-21)!)(0) - (0) + (-1)(-(-1)!)((20)!/(20-(-1))!/(-1)!)*n20-(-1)+(1/2)*n20-0+(1/12)(20!/(20-1)!/1!)*n20-1+(0)n20-2+(-1)(1/120)(20!/(20-3)!/3!)*n20-3+(0)n20-4+(1/252)(20!/(20-5)!/5!)*n20-5+(0)n20-6+(-1)(1/240)(20!/(20-7)!/7!)*n20-7+(0)n20-8+(1/132)(20!/(20-9)!/9!)*n20-9+(0)n20-10+(-1)(691/32760)(20!/(20-11)!/11!)*n20-11+(0)n20-12+(1/12)(20!/(20-13)!/13!)*n20-13+(0)n20-14+(-1)(3617/8160)(20!/(20-15)!/15!)*n20-15+(0)n20-16+(43867/14364)(20!/(20-17)!/17!)*n20-17+(0)n20-18+(-1)(174611/6600)(20!/(20-19)!/19!)*n20-19+(0)n20-20
ζ(-19) = 1+19 + 2+19 + 3+19 + ... + n+19 = π-19 * 2-20 /(-20)! * (±(-20)!)(0) - (-1)(174611/6600) + (-1)(-(-1)!)((19)!/(19-(-1))!/(-1)!)*n19-(-1)+(1/2)*n19-0+(1/12)(19!/(19-1)!/1!)*n19-1+(0)n19-2+(-1)(1/120)(19!/(19-3)!/3!)*n19-3+(0)n19-4+(1/252)(19!/(19-5)!/5!)*n19-5+(0)n19-6+(-1)(1/240)(19!/(19-7)!/7!)*n19-7+(0)n19-8+(1/132)(19!/(19-9)!/9!)*n19-9+(0)n19-10+(-1)(691/32760)(19!/(19-11)!/11!)*n19-11+(0)n19-12+(1/12)(19!/(19-13)!/13!)*n19-13+(0)n19-14+(-1)(3617/8160)(19!/(19-15)!/15!)*n19-15+(0)n19-16+(43867/14364)(19!/(19-17)!/17!)*n19-17+(0)n19-18+(-1)(174611/6600)(19!/(19-19)!/19!)*n19-19
ζ(-18) = 1+18 + 2+18 + 3+18 + ... + n+18 = π-18 * 2-19 /(-19)! * (±(-19)!)(0) - (0) + (-1)(-(-1)!)((18)!/(18-(-1))!/(-1)!)*n18-(-1)+(1/2)*n18-0+(1/12)(18!/(18-1)!/1!)*n18-1+(0)n18-2+(-1)(1/120)(18!/(18-3)!/3!)*n18-3+(0)n18-4+(1/252)(18!/(18-5)!/5!)*n18-5+(0)n18-6+(-1)(1/240)(18!/(18-7)!/7!)*n18-7+(0)n18-8+(1/132)(18!/(18-9)!/9!)*n18-9+(0)n18-10+(-1)(691/32760)(18!/(18-11)!/11!)*n18-11+(0)n18-12+(1/12)(18!/(18-13)!/13!)*n18-13+(0)n18-14+(-1)(3617/8160)(18!/(18-15)!/15!)*n18-15+(0)n18-16+(43867/14364)(18!/(18-17)!/17!)*n18-17+(0)n18-18
ζ(-17) = 1+17 + 2+17 + 3+17 + ... + n+17 = π-17 * 2-18 /(-18)! * (±(-18)!)(0) - (43867/14364) + (-1)(-(-1)!)((17)!/(17-(-1))!/(-1)!)*n17-(-1)+(1/2)*n17-0+(1/12)(17!/(17-1)!/1!)*n17-1+(0)n17-2+(-1)(1/120)(17!/(17-3)!/3!)*n17-3+(0)n17-4+(1/252)(17!/(17-5)!/5!)*n17-5+(0)n17-6+(-1)(1/240)(17!/(17-7)!/7!)*n17-7+(0)n17-8+(1/132)(17!/(17-9)!/9!)*n17-9+(0)n17-10+(-1)(691/32760)(17!/(17-11)!/11!)*n17-11+(0)n17-12+(1/12)(17!/(17-13)!/13!)*n17-13+(0)n17-14+(-1)(3617/8160)(17!/(17-15)!/15!)*n17-15+(0)n17-16+(43867/14364)(17!/(17-17)!/17!)*n17-17
ζ(-16) = 1+16 + 2+16 + 3+16 + ... + n+16 = π-16 * 2-17 /(-17)! * (±(-17)!)(0) - (0) + (-1)(-(-1)!)((16)!/(16-(-1))!/(-1)!)*n16-(-1)+(1/2)*n16-0+(1/12)(16!/(16-1)!/1!)*n16-1+(0)n16-2+(-1)(1/120)(16!/(16-3)!/3!)*n16-3+(0)n16-4+(1/252)(16!/(16-5)!/5!)*n16-5+(0)n16-6+(-1)(1/240)(16!/(16-7)!/7!)*n16-7+(0)n16-8+(1/132)(16!/(16-9)!/9!)*n16-9+(0)n16-10+(-1)(691/32760)(16!/(16-11)!/11!)*n16-11+(0)n16-12+(1/12)(16!/(16-13)!/13!)*n16-13+(0)n16-14+(-1)(3617/8160)(16!/(16-15)!/15!)*n16-15+(0)n16-16
ζ(-15) = 1+15 + 2+15 + 3+15 + ... + n+15 = π-15 * 2-16 /(-16)! * (±(-16)!)(0) - (-1)(3617/8160) + (-1)(-(-1)!)((15)!/(15-(-1))!/(-1)!)*n15-(-1)+(1/2)*n15-0+(1/12)(15!/(15-1)!/1!)*n15-1+(0)n15-2+(-1)(1/120)(15!/(15-3)!/3!)*n15-3+(0)n15-4+(1/252)(15!/(15-5)!/5!)*n15-5+(0)n15-6+(-1)(1/240)(15!/(15-7)!/7!)*n15-7+(0)n15-8+(1/132)(15!/(15-9)!/9!)*n15-9+(0)n15-10+(-1)(691/32760)(15!/(15-11)!/11!)*n15-11+(0)n15-12+(1/12)(15!/(15-13)!/13!)*n15-13+(0)n15-14+(-1)(3617/8160)(15!/(15-15)!/15!)*n15-15
ζ(-14) = 1+14 + 2+14 + 3+14 + ... + n+14 = π-14 * 2-15 /(-15)! * (±(-15)!)(0) - (0) + (-1)(-(-1)!)((14)!/(14-(-1))!/(-1)!)*n14-(-1)+(1/2)*n14-0+(1/12)(14!/(14-1)!/1!)*n14-1+(0)n14-2+(-1)(1/120)(14!/(14-3)!/3!)*n14-3+(0)n14-4+(1/252)(14!/(14-5)!/5!)*n14-5+(0)n14-6+(-1)(1/240)(14!/(14-7)!/7!)*n14-7+(0)n14-8+(1/132)(14!/(14-9)!/9!)*n14-9+(0)n14-10+(-1)(691/32760)(14!/(14-11)!/11!)*n14-11+(0)n14-12+(1/12)(14!/(14-13)!/13!)*n14-13+(0)n14-14
ζ(-13) = 1+13 + 2+13 + 3+13 + ... + n+13 = π-13 * 2-14 /(-14)! * (±(-14)!)(0) - (1/12) + (-1)(-(-1)!)((13)!/(13-(-1))!/(-1)!)*n13-(-1)+(1/2)*n13-0+(1/12)(13!/(13-1)!/1!)*n13-1+(0)n13-2+(-1)(1/120)(13!/(13-3)!/3!)*n13-3+(0)n13-4+(1/252)(13!/(13-5)!/5!)*n13-5+(0)n13-6+(-1)(1/240)(13!/(13-7)!/7!)*n13-7+(0)n13-8+(1/132)(13!/(13-9)!/9!)*n13-9+(0)n13-10+(-1)(691/32760)(13!/(13-11)!/11!)*n13-11+(0)n13-12+(1/12)(13!/(13-13)!/13!)*n13-13
ζ(-12) = 1+12 + 2+12 + 3+12 + ... + n+12 = π-12 * 2-13 /(-13)! * (±(-13)!)(0) - (0) + (-1)(-(-1)!)((12)!/(12-(-1))!/(-1)!)*n12-(-1)+(1/2)*n12-0+(1/12)(12!/(12-1)!/1!)*n12-1+(0)n12-2+(-1)(1/120)(12!/(12-3)!/3!)*n12-3+(0)n12-4+(1/252)(12!/(12-5)!/5!)*n12-5+(0)n12-6+(-1)(1/240)(12!/(12-7)!/7!)*n12-7+(0)n12-8+(1/132)(12!/(12-9)!/9!)*n12-9+(0)n12-10+(-1)(691/32760)(12!/(12-11)!/11!)*n12-11+(0)n12-12 = n(n+1)/2 * (2n+1)/3 * (105n10 +525n9 +525n8 -1050n7 -1190n6 +2310n5 +1420n4 -3285n3 -287n2 +2073n-691)/455
ζ(-11) = 1+11 + 2+11 + 3+11 + ... + n+11 = π-11 * 2-12 /(-12)! * (±(-12)!)(0) - (-1)(691/32760) + (-1)(-(-1)!)((11)!/(11-(-1))!/(-1)!)*n11-(-1)+(1/2)*n11-0+(1/12)(11!/(11-1)!/1!)*n11-1+(0)n11-2+(-1)(1/120)(11!/(11-3)!/3!)*n11-3+(0)n11-4+(1/252)(11!/(11-5)!/5!)*n11-5+(0)n11-6+(-1)(1/240)(11!/(11-7)!/7!)*n11-7+(0)n11-8+(1/132)(11!/(11-9)!/9!)*n11-9+(0)n11-10+(-1)(691/32760)(11!/(11-11)!/11!)*n11-11 = n(n+1)/2 * n(n+1)/2 * (2n8 +8n7 +4n6 -16n5 -5n4 +26n3 -3n2 -20n+10)/6
ζ(-10) = 1+10 + 2+10 + 3+10 + ... + n+10 = π-10 * 2-11 /(-11)! * (±(-11)!)(0) - (0) + (-1)(-(-1)!)((10)!/(10-(-1))!/(-1)!)*n10-(-1)+(1/2)*n10-0+(1/12)(10!/(10-1)!/1!)*n10-1+(0)n10-2+(-1)(1/120)(10!/(10-3)!/3!)*n10-3+(0)n10-4+(1/252)(10!/(10-5)!/5!)*n10-5+(0)n10-6+(-1)(1/240)(10!/(10-7)!/7!)*n10-7+(0)n10-8+(1/132)(10!/(10-9)!/9!)*n10-9+(0)n10-10 = n(n+1)/2 * (2n+1)/3 * (n2 +n-1) * (3n6 +9n5 +2n4 -11n3 +3n2 +10n-5)/11
ζ(-9) = 1+9 + 2+9 + 3+9 + ... + n+9 = π-9 * 2-10 /(-10)! * (±(-10)!)(0) - (1/132) + (-1)(-(-1)!)(( 9)!/( 9-(-1))!/(-1)!)*n 9-(-1)+(1/2)*n 9-0+(1/12)( 9!/( 9-1)!/1!)*n 9-1+(0)n 9-2+(-1)(1/120)( 9!/( 9-3)!/3!)*n 9-3+(0)n 9-4+(1/252)( 9!/( 9-5)!/5!)*n 9-5+(0)n 9-6+(-1)(1/240)( 9!/( 9-7)!/7!)*n 9-7+(0)n 9-8+(1/132)( 9!/( 9-9)!/9!)*n 9-9 = n(n+1)/2 * n(n+1)/2 * (n2 +n-1) * (2n4 +4n3 -n2 -3n+3)/5
ζ(-8) = 1+8 + 2+8 + 3+8 + ... + n+8 = π-8 * 2-9 /(-9 )! * (±(-9 )!)(0) - (0) + (-1)(-(-1)!)(( 8)!/( 8-(-1))!/(-1)!)*n 8-(-1)+(1/2)*n 8-0+(1/12)( 8!/( 8-1)!/1!)*n 8-1+(0)n 8-2+(-1)(1/120)( 8!/( 8-3)!/3!)*n 8-3+(0)n 8-4+(1/252)( 8!/( 8-5)!/5!)*n 8-5+(0)n 8-6+(-1)(1/240)( 8!/( 8-7)!/7!)*n 8-7+(0)n 8-8 = n(n+1)/2 * (2n+1)/3 * (5n6 +15n5 +5n4 -15n3 -n2 +9n-3)/15
ζ(-7) = 1+7 + 2+7 + 3+7 + ... + n+7 = π-7 * 2-8 /(-8 )! * (±(-8 )!)(0) - (-1)(1/240) + (-1)(-(-1)!)(( 7)!/( 7-(-1))!/(-1)!)*n 7-(-1)+(1/2)*n 7-0+(1/12)( 7!/( 7-1)!/1!)*n 7-1+(0)n 7-2+(-1)(1/120)( 7!/( 7-3)!/3!)*n 7-3+(0)n 7-4+(1/252)( 7!/( 7-5)!/5!)*n 7-5+(0)n 7-6+(-1)(1/240)( 7!/( 7-7)!/7!)*n 7-7 = n(n+1)/2 * n(n+1)/2 * (3n4 + 6n3 -n2 -4n-2)/6
ζ(-6) = 1+6 + 2+6 + 3+6 + ... + n+6 = π-6 * 2-7 /(-7 )! * (±(-7 )!)(0) - (0) + (-1)(-(-1)!)(( 6)!/( 6-(-1))!/(-1)!)*n 6-(-1)+(1/2)*n 6-0+(1/12)( 6!/( 6-1)!/1!)*n 6-1+(0)n 6-2+(-1)(1/120)( 6!/( 6-3)!/3!)*n 6-3+(0)n 6-4+(1/252)( 6!/( 6-5)!/5!)*n 6-5+(0)n 6-6 = n(n+1)/2 * (2n+1)/3 * (3n4 + 6n3 -3n+1)/7
ζ(-5) = 1+5 + 2+5 + 3+5 + ... + n+5 = π-5 * 2-6 /(-6 )! * (±(-6 )!)(0) - (1/252) + (-1)(-(-1)!)(( 5)!/( 5-(-1))!/(-1)!)*n 5-(-1)+(1/2)*n 5-0+(1/12)( 5!/( 5-1)!/1!)*n 5-1+(0)n 5-2+(-1)(1/120)( 5!/( 5-3)!/3!)*n 5-3+(0)n 5-4+(1/252)( 5!/( 5-5)!/5!)*n 5-5 = n(n+1)/2 * n(n+1)/2 * (2n2 + 2n-1)/3
ζ(-4) = 1+4 + 2+4 + 3+4 + ... + n+4 = π-4 * 2-5 /(-5 )! * (±(-5 )!)(0) - (0) + (-1)(-(-1)!)(( 4)!/( 4-(-1))!/(-1)!)*n 4-(-1)+(1/2)*n 4-0+(1/12)( 4!/( 4-1)!/1!)*n 4-1+(0)n 4-2+(-1)(1/120)( 4!/( 4-3)!/3!)*n 4-3+(0)n 4-4 = n(n+1)/2 * (2n+1)/3 * (3n2 + 3n-1)/5
ζ(-3) = 1+3 + 2+3 + 3+3 + ... + n+3 = π-3 * 2-4 /(-4 )! * (±(-4 )!)(0) - (-1)(1/120) + (-1)(-(-1)!)(( 3)!/( 3-(-1))!/(-1)!)*n 3-(-1)+(1/2)*n 3-0+(1/12)( 3!/( 3-1)!/1!)*n 3-1+(0)n 3-2+(-1)(1/120)( 3!/( 3-3)!/3!)*n 3-3 = n(n+1)/2 * n(n+1)/2
ζ(-2) = 1+2 + 2+2 + 3+2 + ... + n+2 = π-2 * 2-3 /(-3 )! * (±(-3 )!)(0) - (0) + (-1)(-(-1)!)(( 2)!/( 2-(-1))!/(-1)!)*n 2-(-1)+(1/2)*n 2-0+(1/12)( 2!/( 2-1)!/1!)*n 2-1+(0)n 2-2 = n(n+1)/2 * (2n+1)/3
ζ(-1) = 1+1 + 2+1 + 3+1 + ... + n+1 = π-1 * 2-2 /(-2 )! * (±(-2 )!)(0) - (1/12) + (-1)(-(-1)!)(( 1)!/( 1-(-1))!/(-1)!)*n 1-(-1)+(1/2)*n 1-0+(1/12)( 1!/( 1-1)!/1!)*n 1-1 = n(n+1)/2
ζ( 0) = 1 0 + 2 0 + 3 0 + ... + n 0 = π 0 *(2-1) /(-1 )! * (-(-1 )!) - (0) + (-1)(-(-1)!)(( 0)!/( 0-(-1))!/(-1)!)*n 0-(-1)+(1/2)*n 0-0 = n
ζ(+1) = 1-1 + 2-1 + 3-1 + ... + n-1 = π+1 * 2 0 /( 0 )! * (0) - (-1)(-(-1 )!) + (-1)(-(-1)!)((-1)!/(-1-(-1))!/(-1)!)*n-1-(-1) + ln(n+1)-1/(2n+2)+0.57721566490153286060651209008240243104215933593992359880576723488486772677766467093694 ... A001620
ζ(+2) = 1-2 + 2-2 + 3-2 + ... = π+2 * 2+1 /( 1 )! * (1/12) - (±(-2 )!)(0) = π2 / 6 = 1.64493406684822643647241516664602518921894990120679843773555822937000747040320087383362890061975870 ... A013661
ζ(+3) = 1-3 + 2-3 + 3-3 + ... = π+3 * 2+2 /( 2 )! * (0) - (±(-3 )!)(0) + ??? = π3 / 25.7943501666186840185586365793965132900509523271312260706140213406494349134925061412251 ... A308637 = 1.20205690315959428539973816151144999076498629234049888179227155534183820578631309018645587360933525 ... A002117
ζ(+4) = 1-4 + 2-4 + 3-4 + ... = π+4 * 2+3 /( 3 )! * (1/120) - (±(-4 )!)(0) = π4 / 90 = 1.08232323371113819151600369654116790277475095191872690768297621544412061618696884655690963594169991 ... A013662
ζ(+5) = 1-5 + 2-5 + 3-5 + ... = π+5 * 2+4 /( 4 )! * (0) - (-1)(±(-5 )!)(0) + ??? = π5 / 295.121509929078814295416301676822594619632418745885100174880081881222512573492113833455 ... A309926 = 1.03692775514336992633136548645703416805708091950191281197419267790380358978628148456004310655713333 ... A013663
ζ(+6) = 1-6 + 2-6 + 3-6 + ... = π+6 * 2+5 /( 5 )! * (1/252) - (±(-6 )!)(0) = π6 / 945 = 1.01734306198444913971451792979092052790181749003285356184240866400433218290195789788277397793853517 ... A013664
ζ(+7) = 1-7 + 2-7 + 3-7 + ... = π+7 * 2+6 /( 6 )! * (0) - (±(-7 )!)(0) + ??? = π7 / 2995.28476444062987421457140194123586447237619811128862116034993083589922581051107464452 ... A309927 = 1.00834927738192282683979754984979675959986356056523870641728313657160147831735573534609696891385132 ... A013665
ζ(+8) = 1-8 + 2-8 + 3-8 + ... = π+8 * 2+7 /( 7 )! * (1/240) - (±(-8 )!)(0) = π8 / 9450 = 1.00407735619794433937868523850865246525896079064985002032911020265258295257474881439528723037237197 ... A013666
ζ(+9) = 1-9 + 2-9 + 3-9 + ... = π+9 * 2+8 /( 8 )! * (0) - (-1)(±(-9 )!)(0) + ??? = π9 / 29749.3509504167924732263575439992360954535708605981514652679131630981776684624977358377 ... A309928 = 1.00200839282608221441785276923241206048560585139488875654859661590978505339025839895039306912716958 ... A013667
ζ(+10) = 1-10 + 2-10 + 3-10 + ... = π+10 * 2+9 /( 9 )! * (1/132) - (±(-10)!)(0) = π10 / 93555 = 1.00099457512781808533714595890031901700601953156447751725778899463629146515191295439704196861038565 ... A013668
ζ(+11) = 1-11 + 2-11 + 3-11 + ... = π+11 * 2+10 /( 10)! * (0) - (±(-11)!)(0) + ??? = π11 / 294058.697516635663068056032177491189612189560972448164117512566969938747449053262053487 ... A309929 = 1.00049418860411946455870228252646993646860643575820861711914143610005405979821981470259184302356062 ... A013669
ζ(+12) = 1-12 + 2-12 + 3-12 + ... = π+12 * 2+11 /( 11)! * (691/32760) - (±(-12)!)(0) = π12 * 691 / 638512875 = 1.00024608655330804829863799804773967096041608845800340453304095213325201968194091304904280855190069 ... A013670
ζ(+13) = 1-13 + 2-13 + 3-13 + ... = π+13 * 2+12 /( 12)! * (0) - (-1)(±(-13)!)(0) + ??? = π13 / 2903320.99437496874471612902548598299518022850873348106519286211097791175125276089735094 ... No OEIS = 1.00012271334757848914675183652635739571427510589550984513670267162089672682984420981289271395326813 ... A013671
ζ(+14) = 1-14 + 2-14 + 3-14 + ... = π+14 * 2+13 /( 13)! * (1/12) - (±(-14)!)(0) = π14 * 2 / 18243225 = 1.00006124813505870482925854510513533374748169616915454948275520225286294102317742087665978297199846 ... A013672
ζ(+15) = 1-15 + 2-15 + 3-15 + ... = π+15 * 2+14 /( 14)! * (0) - (±(-15)!)(0) + ??? = π15 / 28657269.3940598590044202589379919803466424134329335109381917049703719697921088276545668 ... No OEIS = 1.00003058823630702049355172851064506258762794870685817750656993289333226715634227957307233434701754 ... A013673
ζ(+16) = 1-16 + 2-16 + 3-16 + ... = π+16 * 2+15 /( 15)! * (3617/8160) - (±(-16)!)(0) = π16 * 3617 / 325641566250 = 1.00001528225940865187173257148763672202323738899047153115310520358878708702795315178628560484632246 ... A013674
ζ(+17) = 1-17 + 2-17 + 3-17 + ... = π+17 * 2+16 /( 16)! * (0) - (-1)(±(-17)!)(0) + ??? = π17 / 282842403.463197426131307236264129094363182272952265735576995225855515620331164084358670 ... No OEIS = 1.00000763719763789976227360029356302921308824909026267909537984397293564329028245934208173863691667 ... A013675
ζ(+18) = 1-18 + 2-18 + 3-18 + ... = π+18 * 2+17 /( 17)! * (43867/14364) - (±(-18)!)(0) = π18 * 43867 / 38979295480125 = 1.00000381729326499983985646164462193973045469721895333114317442998763003954265004563800196866898964 ... A013676
ζ(+19) = 1-19 + 2-19 + 3-19 + ... = π+19 * 2+18 /( 18)! * (0) - (±(-19)!)(0) + ??? = π19 / 2791558622.71018270391989516441857455178217039199704213989473442616757883445034218379359 ... No OEIS = 1.00000190821271655393892565695779510135325857114483863023593304676182394970534130931266422711807630 ... A013677
ζ(+20) = 1-20 + 2-20 + 3-20 + ... = π+20 * 2+19 /( 19)! * (174611/6600) - (±(-20)!)(0) = π20 * 174611 / 1531329465290625 = 1.00000095396203387279611315203868344934594379418741059575005648985113751373114390025783609797638747 ... A013678


Eta Function Table (Advanced Mode)

[Vixra] [PDF] My Email: isaac.mor@hotmail.com

PDF

Here is my Advanced Mode Table:

η(-20) = 1+20 - 2+20 + 3+20 - ... ± n+20 = (0) + (-1)(n-1) * [ (1/2)*n20-0+(1/4)(20!/(20-1)!/1!)*n20-1+(0)n20-2+(-1)(1/8)(20!/(20-3)!/3!)*n20-3+(0)n20-4+(1/4)(20!/(20-5)!/5!)*n20-5+(0)n20-6+(-1)(17/16)(20!/(20-7)!/7!)*n20-7+(0)n20-8+(31/4)(20!/(20-9)!/9!)*n20-9+(0)n20-10+(-1)(691/8)(20!/(20-11)!/11!)*n20-11+(0)n20-12+(5461/4)(20!/(20-13)!/13!)*n20-13+(0)n20-14+(-1)(929569/32)(20!/(20-15)!/15!)*n20-15+(0)n20-16+(3202291/4)(20!/(20-17)!/17!)*n20-17+(0)n20-18+(-1)(221930581/8)(20!/(20-19)!/19!)*n20-19+(0)n20-20 ]
η(-19) = 1+19 - 2+19 + 3+19 - ... ± n+19 = (-1)(221930581/8) + (-1)(n-1) * [ (1/2)*n19-0+(1/4)(19!/(19-1)!/1!)*n19-1+(0)n19-2+(-1)(1/8)(19!/(19-3)!/3!)*n19-3+(0)n19-4+(1/4)(19!/(19-5)!/5!)*n19-5+(0)n19-6+(-1)(17/16)(19!/(19-7)!/7!)*n19-7+(0)n19-8+(31/4)(19!/(19-9)!/9!)*n19-9+(0)n19-10+(-1)(691/8)(19!/(19-11)!/11!)*n19-11+(0)n19-12+(5461/4)(19!/(19-13)!/13!)*n19-13+(0)n19-14+(-1)(929569/32)(19!/(19-15)!/15!)*n19-15+(0)n19-16+(3202291/4)(19!/(19-17)!/17!)*n19-17+(0)n19-18+(-1)(221930581/8)(19!/(19-19)!/19!)*n19-19 ]
η(-18) = 1+18 - 2+18 + 3+18 - ... ± n+18 = (0) + (-1)(n-1) * [ (1/2)*n18-0+(1/4)(18!/(18-1)!/1!)*n18-1+(0)n18-2+(-1)(1/8)(18!/(18-3)!/3!)*n18-3+(0)n18-4+(1/4)(18!/(18-5)!/5!)*n18-5+(0)n18-6+(-1)(17/16)(18!/(18-7)!/7!)*n18-7+(0)n18-8+(31/4)(18!/(18-9)!/9!)*n18-9+(0)n18-10+(-1)(691/8)(18!/(18-11)!/11!)*n18-11+(0)n18-12+(5461/4)(18!/(18-13)!/13!)*n18-13+(0)n18-14+(-1)(929569/32)(18!/(18-15)!/15!)*n18-15+(0)n18-16+(3202291/4)(18!/(18-17)!/17!)*n18-17+(0)n18-18 ]
η(-17) = 1+17 - 2+17 + 3+17 - ... ± n+17 = (3202291/4) + (-1)(n-1) * [ (1/2)*n17-0+(1/4)(17!/(17-1)!/1!)*n17-1+(0)n17-2+(-1)(1/8)(17!/(17-3)!/3!)*n17-3+(0)n17-4+(1/4)(17!/(17-5)!/5!)*n17-5+(0)n17-6+(-1)(17/16)(17!/(17-7)!/7!)*n17-7+(0)n17-8+(31/4)(17!/(17-9)!/9!)*n17-9+(0)n17-10+(-1)(691/8)(17!/(17-11)!/11!)*n17-11+(0)n17-12+(5461/4)(17!/(17-13)!/13!)*n17-13+(0)n17-14+(-1)(929569/32)(17!/(17-15)!/15!)*n17-15+(0)n17-16+(3202291/4)(17!/(17-17)!/17!)*n17-17 ]
η(-16) = 1+16 - 2+16 + 3+16 - ... ± n+16 = (0) + (-1)(n-1) * [ (1/2)*n16-0+(1/4)(16!/(16-1)!/1!)*n16-1+(0)n16-2+(-1)(1/8)(16!/(16-3)!/3!)*n16-3+(0)n16-4+(1/4)(16!/(16-5)!/5!)*n16-5+(0)n16-6+(-1)(17/16)(16!/(16-7)!/7!)*n16-7+(0)n16-8+(31/4)(16!/(16-9)!/9!)*n16-9+(0)n16-10+(-1)(691/8)(16!/(16-11)!/11!)*n16-11+(0)n16-12+(5461/4)(16!/(16-13)!/13!)*n16-13+(0)n16-14+(-1)(929569/32)(16!/(16-15)!/15!)*n16-15+(0)n16-16 ]
η(-15) = 1+15 - 2+15 + 3+15 - ... ± n+15 = (-1)(929569/32) + (-1)(n-1) * [ (1/2)*n15-0+(1/4)(15!/(15-1)!/1!)*n15-1+(0)n15-2+(-1)(1/8)(15!/(15-3)!/3!)*n15-3+(0)n15-4+(1/4)(15!/(15-5)!/5!)*n15-5+(0)n15-6+(-1)(17/16)(15!/(15-7)!/7!)*n15-7+(0)n15-8+(31/4)(15!/(15-9)!/9!)*n15-9+(0)n15-10+(-1)(691/8)(15!/(15-11)!/11!)*n15-11+(0)n15-12+(5461/4)(15!/(15-13)!/13!)*n15-13+(0)n15-14+(-1)(929569/32)(15!/(15-15)!/15!)*n15-15 ]
η(-14) = 1+14 - 2+14 + 3+14 - ... ± n+14 = (0) + (-1)(n-1) * [ (1/2)*n14-0+(1/4)(14!/(14-1)!/1!)*n14-1+(0)n14-2+(-1)(1/8)(14!/(14-3)!/3!)*n14-3+(0)n14-4+(1/4)(14!/(14-5)!/5!)*n14-5+(0)n14-6+(-1)(17/16)(14!/(14-7)!/7!)*n14-7+(0)n14-8+(31/4)(14!/(14-9)!/9!)*n14-9+(0)n14-10+(-1)(691/8)(14!/(14-11)!/11!)*n14-11+(0)n14-12+(5461/4)(14!/(14-13)!/13!)*n14-13+(0)n14-14 ]
η(-13) = 1+13 - 2+13 + 3+13 - ... ± n+13 = (5461/4) + (-1)(n-1) * [ (1/2)*n13-0+(1/4)(13!/(13-1)!/1!)*n13-1+(0)n13-2+(-1)(1/8)(13!/(13-3)!/3!)*n13-3+(0)n13-4+(1/4)(13!/(13-5)!/5!)*n13-5+(0)n13-6+(-1)(17/16)(13!/(13-7)!/7!)*n13-7+(0)n13-8+(31/4)(13!/(13-9)!/9!)*n13-9+(0)n13-10+(-1)(691/8)(13!/(13-11)!/11!)*n13-11+(0)n13-12+(5461/4)(13!/(13-13)!/13!)*n13-13 ]
η(-12) = 1+12 - 2+12 + 3+12 - ... ± n+12 = (0) + (-1)(n-1) * [ (1/2)*n12-0+(1/4)(12!/(12-1)!/1!)*n12-1+(0)n12-2+(-1)(1/8)(12!/(12-3)!/3!)*n12-3+(0)n12-4+(1/4)(12!/(12-5)!/5!)*n12-5+(0)n12-6+(-1)(17/16)(12!/(12-7)!/7!)*n12-7+(0)n12-8+(31/4)(12!/(12-9)!/9!)*n12-9+(0)n12-10+(-1)(691/8)(12!/(12-11)!/11!)*n12-11+(0)n12-12 ]
η(-11) = 1+11 - 2+11 + 3+11 - ... ± n+11 = (-1)(691/8) + (-1)(n-1) * [ (1/2)*n11-0+(1/4)(11!/(11-1)!/1!)*n11-1+(0)n11-2+(-1)(1/8)(11!/(11-3)!/3!)*n11-3+(0)n11-4+(1/4)(11!/(11-5)!/5!)*n11-5+(0)n11-6+(-1)(17/16)(11!/(11-7)!/7!)*n11-7+(0)n11-8+(31/4)(11!/(11-9)!/9!)*n11-9+(0)n11-10+(-1)(691/8)(11!/(11-11)!/11!)*n11-11 ]
η(-10) = 1+10 - 2+10 + 3+10 - ... ± n+10 = (0) + (-1)(n-1) * [ (1/2)*n10-0+(1/4)(10!/(10-1)!/1!)*n10-1+(0)n10-2+(-1)(1/8)(10!/(10-3)!/3!)*n10-3+(0)n10-4+(1/4)(10!/(10-5)!/5!)*n10-5+(0)n10-6+(-1)(17/16)(10!/(10-7)!/7!)*n10-7+(0)n10-8+(31/4)(10!/(10-9)!/9!)*n10-9+(0)n10-10 ]
η(-9) = 1+9 - 2+9 + 3+9 - ... ± n+9 = (31/4) + (-1)(n-1) * [ (1/2)*n 9-0+(1/4)( 9!/( 9-1)!/1!)*n 9-1+(0)n 9-2+(-1)(1/8)( 9!/( 9-3)!/3!)*n 9-3+(0)n 9-4+(1/4)( 9!/( 9-5)!/5!)*n 9-5+(0)n 9-6+(-1)(17/16)( 9!/( 9-7)!/7!)*n 9-7+(0)n 9-8+(31/4)( 9!/(10-9)!/9!)*n 9-9 ]
η(-8) = 1+8 - 2+8 + 3+8 - ... ± n+8 = (0) + (-1)(n-1) * [ (1/2)*n 8-0+(1/4)( 8!/( 8-1)!/1!)*n 8-1+(0)n 8-2+(-1)(1/8)( 8!/( 8-3)!/3!)*n 8-3+(0)n 8-4+(1/4)( 8!/( 8-5)!/5!)*n 8-5+(0)n 8-6+(-1)(17/16)( 8!/( 8-7)!/7!)*n 8-7+(0)n 8-8 ]
η(-7) = 1+7 - 2+7 + 3+7 - ... ± n+7 = (-1)(17/16) + (-1)(n-1) * [ (1/2)*n 7-0+(1/4)( 7!/( 7-1)!/1!)*n 7-1+(0)n 7-2+(-1)(1/8)( 7!/( 7-3)!/3!)*n 7-3+(0)n 7-4+(1/4)( 7!/( 7-5)!/5!)*n 7-5+(0)n 7-6+(-1)(17/16)( 7!/( 7-7)!/7!)*n 7-7 ]
η(-6) = 1+6 - 2+6 + 3+6 - ... ± n+6 = (0) + (-1)(n-1) * [ (1/2)*n 6-0+(1/4)( 6!/( 6-1)!/1!)*n 6-1+(0)n 6-2+(-1)(1/8)( 6!/( 6-3)!/3!)*n 6-3+(0)n 6-4+(1/4)( 6!/( 6-5)!/5!)*n 6-5+(0)n 6-6 ]
η(-5) = 1+5 - 2+5 + 3+5 - ... ± n+5 = (1/4) + (-1)(n-1) * [ (1/2)*n 5-0+(1/4)( 5!/( 5-1)!/1!)*n 5-1+(0)n 5-2+(-1)(1/8)( 5!/( 5-3)!/3!)*n 5-3+(0)n 5-4+(1/4)( 5!/( 5-5)!/5!)*n 5-5 ]
η(-4) = 1+4 - 2+4 + 3+4 - ... ± n+4 = (0) + (-1)(n-1) * [ (1/2)*n 4-0+(1/4)( 4!/( 4-1)!/1!)*n 4-1+(0)n 4-2+(-1)(1/8)( 4!/( 4-3)!/3!)*n 4-3+(0)n 4-4 ]
η(-3) = 1+3 - 2+3 + 3+3 - ... ± n+3 = (-1)(1/8) + (-1)(n-1) * [ (1/2)*n 3-0+(1/4)( 3!/( 3-1)!/1!)*n 3-1+(0)n 3-2+(-1)(1/8)( 3!/( 3-3)!/3!)*n 3-3 ]
η(-2) = 1+2 - 2+2 + 3+2 - ... ± n+2 = (0) + (-1)(n-1) * [ (1/2)*n 2-0+(1/4)( 2!/( 2-1)!/1!)*n 2-1+(0)n 2-2 ]
η(-1) = 1+1 - 2+1 + 3+1 - ... ± n+1 = (1/4) + (-1)(n-1) * [ (1/2)*n 1-0+(1/4)( 1!/( 1-1)!/1!)*n 1-1 ]
η( 0) = 1 0 - 2 0 + 3 0 - ... ± n 0 = (1/2) + (-1)(n-1) * [ (1/2)*n 0-0 ]
η(+1) = 1-1 - 2-1 + 3-1 - ... = π+1 * (1-2 0 )/(1-2+1 ) /( 0 )! * (1/2) + ln(2)
η(+2) = 1-2 - 2-2 + 3-2 - ... = π+2 * (1-2+1 )/(1-2+2 ) /( 1 )! * (1/4)
η(+3) = 1-3 - 2-3 + 3-3 - ... = π+3 * (1-2+2 )/(1-2+3 ) /( 2 )! * (0) + ?????
η(+4) = 1-4 - 2-4 + 3-4 - ... = π+4 * (1-2+3 )/(1-2+4 ) /( 3 )! * (1/8)
η(+5) = 1-5 - 2-5 + 3-5 - ... = π+5 * (1-2+4 )/(1-2+5 ) /( 4 )! * (0) + ?????
η(+6) = 1-6 - 2-6 + 3-6 - ... = π+6 * (1-2+5 )/(1-2+6 ) /( 5 )! * (1/4)
η(+7) = 1-7 - 2-7 + 3-7 - ... = π+7 * (1-2+6 )/(1-2+7 ) /( 6 )! * (0) + ?????
η(+8) = 1-8 - 2-8 + 3-8 - ... = π+8 * (1-2+7 )/(1-2+8 ) /( 7 )! * (17/16)
η(+9) = 1-9 - 2-9 + 3-9 - ... = π+9 * (1-2+8 )/(1-2+9 ) /( 8 )! * (0) + ?????
η(+10) = 1-10 - 2-10 + 3-10 - ... = π+10 * (1-2+9 )/(1-2+10) /( 9 )! * (31/4)
η(+11) = 1-11 - 2-11 + 3-11 - ... = π+11 * (1-2+10)/(1-2+11) /( 10)! * (0) + ?????
η(+12) = 1-12 - 2-12 + 3-12 - ... = π+12 * (1-2+11)/(1-2+12) /( 11)! * (691/8)
η(+13) = 1-13 - 2-13 + 3-13 - ... = π+13 * (1-2+12)/(1-2+13) /( 12)! * (0) + ?????
η(+14) = 1-14 - 2-14 + 3-14 - ... = π+14 * (1-2+13)/(1-2+14) /( 13)! * (5461/4)
η(+15) = 1-15 - 2-15 + 3-15 - ... = π+15 * (1-2+14)/(1-2+15) /( 14)! * (0) + ?????
η(+16) = 1-16 - 2-16 + 3-16 - ... = π+16 * (1-2+15)/(1-2+16) /( 15)! * (929569/32)
η(+17) = 1-17 - 2-17 + 3-17 - ... = π+17 * (1-2+16)/(1-2+17) /( 16)! * (0) + ?????
η(+18) = 1-18 - 2-18 + 3-18 - ... = π+18 * (1-2+17)/(1-2+18) /( 17)! * (3202291/4)
η(+19) = 1-19 - 2-19 + 3-19 - ... = π+19 * (1-2+18)/(1-2+19) /( 18)! * (0) + ?????
η(+20) = 1-20 - 2-20 + 3-20 - ... = π+20 * (1-2+19)/(1-2+20) /( 19)! * (221930581/8)


Eta Function Table (Experimental Mode)

Here is my Experimental Mode Table:

η(-20) = 1+20 - 2+20 + 3+20 - ... ± n+20 = π-20 * (1-2-21)/(1-2-20) /(-21)! * (±(-21)!)(0) + (0) + (-1)(±(-1)!)(0)(20!/(20-(-1))!/(-1)!)*n20-(-1) + (-1)(n-1) * [ (1/2)*n20-0+(1/4)(20!/(20-1)!/1!)*n20-1+(0)n20-2+(-1)(1/8)(20!/(20-3)!/3!)*n20-3+(0)n20-4+(1/4)(20!/(20-5)!/5!)*n20-5+(0)n20-6+(-1)(17/16)(20!/(20-7)!/7!)*n20-7+(0)n20-8+(31/4)(20!/(20-9)!/9!)*n20-9+(0)n20-10+(-1)(691/8)(20!/(20-11)!/11!)*n20-11+(0)n20-12+(5461/4)(20!/(20-13)!/13!)*n20-13+(0)n20-14+(-1)(929569/32)(20!/(20-15)!/15!)*n20-15+(0)n20-16+(3202291/4)(20!/(20-17)!/17!)*n20-17+(0)n20-18+(-1)(221930581/8)(20!/(20-19)!/19!)*n20-19+(0)n20-20 ]
η(-19) = 1+19 - 2+19 + 3+19 - ... ± n+19 = π-19 * (1-2-20)/(1-2-19) /(-20)! * (±(-20)!)(0) + (-1)(221930581/8) + (-1)(±(-1)!)(0)(19!/(19-(-1))!/(-1)!)*n19-(-1) + (-1)(n-1) * [ (1/2)*n19-0+(1/4)(19!/(19-1)!/1!)*n19-1+(0)n19-2+(-1)(1/8)(19!/(19-3)!/3!)*n19-3+(0)n19-4+(1/4)(19!/(19-5)!/5!)*n19-5+(0)n19-6+(-1)(17/16)(19!/(19-7)!/7!)*n19-7+(0)n19-8+(31/4)(19!/(19-9)!/9!)*n19-9+(0)n19-10+(-1)(691/8)(19!/(19-11)!/11!)*n19-11+(0)n19-12+(5461/4)(19!/(19-13)!/13!)*n19-13+(0)n19-14+(-1)(929569/32)(19!/(19-15)!/15!)*n19-15+(0)n19-16+(3202291/4)(19!/(19-17)!/17!)*n19-17+(0)n19-18+(-1)(221930581/8)(19!/(19-19)!/19!)*n19-19 ]
η(-18) = 1+18 - 2+18 + 3+18 - ... ± n+18 = π-18 * (1-2-19)/(1-2-18) /(-19)! * (±(-19)!)(0) + (0) + (-1)(±(-1)!)(0)(18!/(18-(-1))!/(-1)!)*n18-(-1) + (-1)(n-1) * [ (1/2)*n18-0+(1/4)(18!/(18-1)!/1!)*n18-1+(0)n18-2+(-1)(1/8)(18!/(18-3)!/3!)*n18-3+(0)n18-4+(1/4)(18!/(18-5)!/5!)*n18-5+(0)n18-6+(-1)(17/16)(18!/(18-7)!/7!)*n18-7+(0)n18-8+(31/4)(18!/(18-9)!/9!)*n18-9+(0)n18-10+(-1)(691/8)(18!/(18-11)!/11!)*n18-11+(0)n18-12+(5461/4)(18!/(18-13)!/13!)*n18-13+(0)n18-14+(-1)(929569/32)(18!/(18-15)!/15!)*n18-15+(0)n18-16+(3202291/4)(18!/(18-17)!/17!)*n18-17+(0)n18-18 ]
η(-17) = 1+17 - 2+17 + 3+17 - ... ± n+17 = π-17 * (1-2-18)/(1-2-17) /(-18)! * (±(-18)!)(0) + (3202291/4) + (-1)(±(-1)!)(0)(17!/(17-(-1))!/(-1)!)*n17-(-1) + (-1)(n-1) * [ (1/2)*n17-0+(1/4)(17!/(17-1)!/1!)*n17-1+(0)n17-2+(-1)(1/8)(17!/(17-3)!/3!)*n17-3+(0)n17-4+(1/4)(17!/(17-5)!/5!)*n17-5+(0)n17-6+(-1)(17/16)(17!/(17-7)!/7!)*n17-7+(0)n17-8+(31/4)(17!/(17-9)!/9!)*n17-9+(0)n17-10+(-1)(691/8)(17!/(17-11)!/11!)*n17-11+(0)n17-12+(5461/4)(17!/(17-13)!/13!)*n17-13+(0)n17-14+(-1)(929569/32)(17!/(17-15)!/15!)*n17-15+(0)n17-16+(3202291/4)(17!/(17-17)!/17!)*n17-17 ]
η(-16) = 1+16 - 2+16 + 3+16 - ... ± n+16 = π-16 * (1-2-17)/(1-2-16) /(-17)! * (±(-17)!)(0) + (0) + (-1)(±(-1)!)(0)(16!/(16-(-1))!/(-1)!)*n16-(-1) + (-1)(n-1) * [ (1/2)*n16-0+(1/4)(16!/(16-1)!/1!)*n16-1+(0)n16-2+(-1)(1/8)(16!/(16-3)!/3!)*n16-3+(0)n16-4+(1/4)(16!/(16-5)!/5!)*n16-5+(0)n16-6+(-1)(17/16)(16!/(16-7)!/7!)*n16-7+(0)n16-8+(31/4)(16!/(16-9)!/9!)*n16-9+(0)n16-10+(-1)(691/8)(16!/(16-11)!/11!)*n16-11+(0)n16-12+(5461/4)(16!/(16-13)!/13!)*n16-13+(0)n16-14+(-1)(929569/32)(16!/(16-15)!/15!)*n16-15+(0)n16-16 ]
η(-15) = 1+15 - 2+15 + 3+15 - ... ± n+15 = π-15 * (1-2-16)/(1-2-15) /(-16)! * (±(-16)!)(0) + (-1)(929569/32) + (-1)(±(-1)!)(0)(15!/(15-(-1))!/(-1)!)*n15-(-1) + (-1)(n-1) * [ (1/2)*n15-0+(1/4)(15!/(15-1)!/1!)*n15-1+(0)n15-2+(-1)(1/8)(15!/(15-3)!/3!)*n15-3+(0)n15-4+(1/4)(15!/(15-5)!/5!)*n15-5+(0)n15-6+(-1)(17/16)(15!/(15-7)!/7!)*n15-7+(0)n15-8+(31/4)(15!/(15-9)!/9!)*n15-9+(0)n15-10+(-1)(691/8)(15!/(15-11)!/11!)*n15-11+(0)n15-12+(5461/4)(15!/(15-13)!/13!)*n15-13+(0)n15-14+(-1)(929569/32)(15!/(15-15)!/15!)*n15-15 ]
η(-14) = 1+14 - 2+14 + 3+14 - ... ± n+14 = π-14 * (1-2-15)/(1-2-14) /(-15)! * (±(-15)!)(0) + (0) + (-1)(±(-1)!)(0)(14!/(14-(-1))!/(-1)!)*n14-(-1) + (-1)(n-1) * [ (1/2)*n14-0+(1/4)(14!/(14-1)!/1!)*n14-1+(0)n14-2+(-1)(1/8)(14!/(14-3)!/3!)*n14-3+(0)n14-4+(1/4)(14!/(14-5)!/5!)*n14-5+(0)n14-6+(-1)(17/16)(14!/(14-7)!/7!)*n14-7+(0)n14-8+(31/4)(14!/(14-9)!/9!)*n14-9+(0)n14-10+(-1)(691/8)(14!/(14-11)!/11!)*n14-11+(0)n14-12+(5461/4)(14!/(14-13)!/13!)*n14-13+(0)n14-14 ]
η(-13) = 1+13 - 2+13 + 3+13 - ... ± n+13 = π-13 * (1-2-14)/(1-2-13) /(-14)! * (±(-14)!)(0) + (5461/4) + (-1)(±(-1)!)(0)(13!/(13-(-1))!/(-1)!)*n13-(-1) + (-1)(n-1) * [ (1/2)*n13-0+(1/4)(13!/(13-1)!/1!)*n13-1+(0)n13-2+(-1)(1/8)(13!/(13-3)!/3!)*n13-3+(0)n13-4+(1/4)(13!/(13-5)!/5!)*n13-5+(0)n13-6+(-1)(17/16)(13!/(13-7)!/7!)*n13-7+(0)n13-8+(31/4)(13!/(13-9)!/9!)*n13-9+(0)n13-10+(-1)(691/8)(13!/(13-11)!/11!)*n13-11+(0)n13-12+(5461/4)(13!/(13-13)!/13!)*n13-13 ]
η(-12) = 1+12 - 2+12 + 3+12 - ... ± n+12 = π-12 * (1-2-13)/(1-2-12) /(-13)! * (±(-13)!)(0) + (0) + (-1)(±(-1)!)(0)(12!/(12-(-1))!/(-1)!)*n12-(-1) + (-1)(n-1) * [ (1/2)*n12-0+(1/4)(12!/(12-1)!/1!)*n12-1+(0)n12-2+(-1)(1/8)(12!/(12-3)!/3!)*n12-3+(0)n12-4+(1/4)(12!/(12-5)!/5!)*n12-5+(0)n12-6+(-1)(17/16)(12!/(12-7)!/7!)*n12-7+(0)n12-8+(31/4)(12!/(12-9)!/9!)*n12-9+(0)n12-10+(-1)(691/8)(12!/(12-11)!/11!)*n12-11+(0)n12-12 ]
η(-11) = 1+11 - 2+11 + 3+11 - ... ± n+11 = π-11 * (1-2-12)/(1-2-11) /(-12)! * (±(-12)!)(0) + (-1)(691/8) + (-1)(±(-1)!)(0)(11!/(11-(-1))!/(-1)!)*n11-(-1) + (-1)(n-1) * [ (1/2)*n11-0+(1/4)(11!/(11-1)!/1!)*n11-1+(0)n11-2+(-1)(1/8)(11!/(11-3)!/3!)*n11-3+(0)n11-4+(1/4)(11!/(11-5)!/5!)*n11-5+(0)n11-6+(-1)(17/16)(11!/(11-7)!/7!)*n11-7+(0)n11-8+(31/4)(11!/(11-9)!/9!)*n11-9+(0)n11-10+(-1)(691/8)(11!/(11-11)!/11!)*n11-11 ]
η(-10) = 1+10 - 2+10 + 3+10 - ... ± n+10 = π-10 * (1-2-11)/(1-2-10) /(-11)! * (±(-11)!)(0) + (0) + (-1)(±(-1)!)(0)(10!/(10-(-1))!/(-1)!)*n10-(-1) + (-1)(n-1) * [ (1/2)*n10-0+(1/4)(10!/(10-1)!/1!)*n10-1+(0)n10-2+(-1)(1/8)(10!/(10-3)!/3!)*n10-3+(0)n10-4+(1/4)(10!/(10-5)!/5!)*n10-5+(0)n10-6+(-1)(17/16)(10!/(10-7)!/7!)*n10-7+(0)n10-8+(31/4)(10!/(10-9)!/9!)*n10-9+(0)n10-10 ]
η(-9) = 1+9 - 2+9 + 3+9 - ... ± n+9 = π-9 * (1-2-10)/(1-2-9 ) /(-10)! * (±(-10)!)(0) + (31/4) + (-1)(±(-1)!)(0)( 9!/( 9-(-1))!/(-1)!)*n 9-(-1) + (-1)(n-1) * [ (1/2)*n 9-0+(1/4)( 9!/( 9-1)!/1!)*n 9-1+(0)n 9-2+(-1)(1/8)( 9!/( 9-3)!/3!)*n 9-3+(0)n 9-4+(1/4)( 9!/( 9-5)!/5!)*n 9-5+(0)n 9-6+(-1)(17/16)( 9!/( 9-7)!/7!)*n 9-7+(0)n 9-8+(31/4)( 9!/(10-9)!/9!)*n 9-9 ]
η(-8) = 1+8 - 2+8 + 3+8 - ... ± n+8 = π-8 * (1-2-9 )/(1-2-8 ) /(-9 )! * (±(-9 )!)(0) + (0) + (-1)(±(-1)!)(0)( 8!/( 8-(-1))!/(-1)!)*n 8-(-1) + (-1)(n-1) * [ (1/2)*n 8-0+(1/4)( 8!/( 8-1)!/1!)*n 8-1+(0)n 8-2+(-1)(1/8)( 8!/( 8-3)!/3!)*n 8-3+(0)n 8-4+(1/4)( 8!/( 8-5)!/5!)*n 8-5+(0)n 8-6+(-1)(17/16)( 8!/( 8-7)!/7!)*n 8-7+(0)n 8-8 ]
η(-7) = 1+7 - 2+7 + 3+7 - ... ± n+7 = π-7 * (1-2-8 )/(1-2-7 ) /(-8 )! * (±(-8 )!)(0) + (-1)(17/16) + (-1)(±(-1)!)(0)( 7!/( 7-(-1))!/(-1)!)*n 7-(-1) + (-1)(n-1) * [ (1/2)*n 7-0+(1/4)( 7!/( 7-1)!/1!)*n 7-1+(0)n 7-2+(-1)(1/8)( 7!/( 7-3)!/3!)*n 7-3+(0)n 7-4+(1/4)( 7!/( 7-5)!/5!)*n 7-5+(0)n 7-6+(-1)(17/16)( 7!/( 7-7)!/7!)*n 7-7 ]
η(-6) = 1+6 - 2+6 + 3+6 - ... ± n+6 = π-6 * (1-2-7 )/(1-2-6 ) /(-7 )! * (±(-7 )!)(0) + (0) + (-1)(±(-1)!)(0)( 6!/( 6-(-1))!/(-1)!)*n 6-(-1) + (-1)(n-1) * [ (1/2)*n 6-0+(1/4)( 6!/( 6-1)!/1!)*n 6-1+(0)n 6-2+(-1)(1/8)( 6!/( 6-3)!/3!)*n 6-3+(0)n 6-4+(1/4)( 6!/( 6-5)!/5!)*n 6-5+(0)n 6-6 ]
η(-5) = 1+5 - 2+5 + 3+5 - ... ± n+5 = π-5 * (1-2-6 )/(1-2-5 ) /(-6 )! * (±(-6 )!)(0) + (1/4) + (-1)(±(-1)!)(0)( 5!/( 5-(-1))!/(-1)!)*n 5-(-1) + (-1)(n-1) * [ (1/2)*n 5-0+(1/4)( 5!/( 5-1)!/1!)*n 5-1+(0)n 5-2+(-1)(1/8)( 5!/( 5-3)!/3!)*n 5-3+(0)n 5-4+(1/4)( 5!/( 5-5)!/5!)*n 5-5 ]
η(-4) = 1+4 - 2+4 + 3+4 - ... ± n+4 = π-4 * (1-2-5 )/(1-2-4 ) /(-5 )! * (±(-5 )!)(0) + (0) + (-1)(±(-1)!)(0)( 4!/( 4-(-1))!/(-1)!)*n 4-(-1) + (-1)(n-1) * [ (1/2)*n 4-0+(1/4)( 4!/( 4-1)!/1!)*n 4-1+(0)n 4-2+(-1)(1/8)( 4!/( 4-3)!/3!)*n 4-3+(0)n 4-4 ]
η(-3) = 1+3 - 2+3 + 3+3 - ... ± n+3 = π-3 * (1-2-4 )/(1-2-3 ) /(-4 )! * (±(-4 )!)(0) + (-1)(1/8) + (-1)(±(-1)!)(0)( 3!/( 3-(-1))!/(-1)!)*n 3-(-1) + (-1)(n-1) * [ (1/2)*n 3-0+(1/4)( 3!/( 3-1)!/1!)*n 3-1+(0)n 3-2+(-1)(1/8)( 3!/( 3-3)!/3!)*n 3-3 ]
η(-2) = 1+2 - 2+2 + 3+2 - ... ± n+2 = π-2 * (1-2-3 )/(1-2-2 ) /(-3 )! * (±(-3 )!)(0) + (0) + (-1)(±(-1)!)(0)( 2!/( 2-(-1))!/(-1)!)*n 2-(-1) + (-1)(n-1) * [ (1/2)*n 2-0+(1/4)( 2!/( 2-1)!/1!)*n 2-1+(0)n 2-2 ]
η(-1) = 1+1 - 2+1 + 3+1 - ... ± n+1 = π-1 * (1-2-2 )/(1-2-1 ) /(-2 )! * (±(-2 )!)(0) + (1/4) + (-1)(±(-1)!)(0)( 1!/( 1-(-1))!/(-1)!)*n 1-(-1) + (-1)(n-1) * [ (1/2)*n 1-0+(1/4)( 1!/( 1-1)!/1!)*n 1-1 ]
η( 0) = 1 0 - 2 0 + 3 0 - ... ± n 0 = π 0 * (1-2-1 )/(1-2 0 ) /(-1 )! * (±(-1 )!)(0) + (0) + (-1)(±(-1)!)(0)( 0!/( 0-(-1))!/(-1)!)*n 0-(-1) + (-1)(n-1) * [ (1/2)*n 0-0 ]
η(+1) = 1-1 - 2-1 + 3-1 - ... = π+1 * (1-2 0 )/(1-2+1 ) /( 0 )! * (0) + (-1)(±(-1 )!)(0) + ln(2)
η(+2) = 1-2 - 2-2 + 3-2 - ... = π+2 * (1-2+1 )/(1-2+2 ) /( 1 )! * (1/4) + (±(-2 )!)(0)
η(+3) = 1-3 - 2-3 + 3-3 - ... = π+3 * (1-2+2 )/(1-2+3 ) /( 2 )! * (0) + (±(-3 )!)(0) + ?????
η(+4) = 1-4 - 2-4 + 3-4 - ... = π+4 * (1-2+3 )/(1-2+4 ) /( 3 )! * (1/8) + (±(-4 )!)(0)
η(+5) = 1-5 - 2-5 + 3-5 - ... = π+5 * (1-2+4 )/(1-2+5 ) /( 4 )! * (0) + (-1)(±(-5 )!)(0) + ?????
η(+6) = 1-6 - 2-6 + 3-6 - ... = π+6 * (1-2+5 )/(1-2+6 ) /( 5 )! * (1/4) + (±(-6 )!)(0)
η(+7) = 1-7 - 2-7 + 3-7 - ... = π+7 * (1-2+6 )/(1-2+7 ) /( 6 )! * (0) + (±(-7 )!)(0) + ?????
η(+8) = 1-8 - 2-8 + 3-8 - ... = π+8 * (1-2+7 )/(1-2+8 ) /( 7 )! * (17/16) + (±(-8 )!)(0)
η(+9) = 1-9 - 2-9 + 3-9 - ... = π+9 * (1-2+8 )/(1-2+9 ) /( 8 )! * (0) + (-1)(±(-9 )!)(0) + ?????
η(+10) = 1-10 - 2-10 + 3-10 - ... = π+10 * (1-2+9 )/(1-2+10) /( 9 )! * (31/4) + (±(-10)!)(0)
η(+11) = 1-11 - 2-11 + 3-11 - ... = π+11 * (1-2+10)/(1-2+11) /( 10)! * (0) + (±(-11)!)(0) + ?????
η(+12) = 1-12 - 2-12 + 3-12 - ... = π+12 * (1-2+11)/(1-2+12) /( 11)! * (691/8) + (±(-12)!)(0)
η(+13) = 1-13 - 2-13 + 3-13 - ... = π+13 * (1-2+12)/(1-2+13) /( 12)! * (0) + (-1)(±(-13)!)(0) + ?????
η(+14) = 1-14 - 2-14 + 3-14 - ... = π+14 * (1-2+13)/(1-2+14) /( 13)! * (5461/4) + (±(-14)!)(0)
η(+15) = 1-15 - 2-15 + 3-15 - ... = π+15 * (1-2+14)/(1-2+15) /( 14)! * (0) + (±(-15)!)(0) + ?????
η(+16) = 1-16 - 2-16 + 3-16 - ... = π+16 * (1-2+15)/(1-2+16) /( 15)! * (929569/32) + (±(-16)!)(0)
η(+17) = 1-17 - 2-17 + 3-17 - ... = π+17 * (1-2+16)/(1-2+17) /( 16)! * (0) + (-1)(±(-17)!)(0) + ?????
η(+18) = 1-18 - 2-18 + 3-18 - ... = π+18 * (1-2+17)/(1-2+18) /( 17)! * (3202291/4) + (±(-18)!)(0)
η(+19) = 1-19 - 2-19 + 3-19 - ... = π+19 * (1-2+18)/(1-2+19) /( 18)! * (0) + (±(-19)!)(0) + ?????
η(+20) = 1-20 - 2-20 + 3-20 - ... = π+20 * (1-2+19)/(1-2+20) /( 19)! * (221930581/8) + (±(-20)!)(0)


Important Note: i know the line at η( 0) could be a controversial step but it seems to fit

its the same idea i used for zeta function when the value there gave -1/2 and now it gives 1/2


Picturing the Zeta Function on the Complex plane

1.1_i1.8

A Deeper Understanding : “origin points”

Many people are using the term “Assigned Value” or “Analytic Continuation” for divergent series
But this explanation is so lacking and can be replaced with a much easier and simpler term of explanation

For me (as I see it) when I am looking at the zeta function I dont see (or use) the term “Assigned Value” or “Analytic Continuation”
Instead I see “spirals” all around the grid!

The simplest way is to first look at the Complex plane ζ(s)=ζ(x+iy)=a+ib where Re(s)>1 and the behavior of convergent points (above spiral picture!)
The spiral swirls around inwards to an unique point which the series Converges - Same goes for the other way around!

When I look at the Complex plane ζ(s)=ζ(x+iy)=a+ib where Re(s)<1 and the behavior of divergent points
The spiral swirls around outwards but if you look closely you will notice that the spiral has a “center point” or an “origin”
and that “origin” is the “Assigned Value” everyone is talking about

when I first started to read about the zeta function I didn’t know what are those “Assigned Values” or “Analytic Continuation”
and how and why people are trying to give a value for divergent series And why that specific value and not something else?
I wanted an explanation other then “because the formula says so” and without going deeper into all the “Analytic Continuation stuff".

Those “origin points” did the trick!

the simplest origin point to understend is η(-1)=1-2+3-4+5-6+...
Origin_Point

the (Assigned) value 1/4 is not the summation of η(-1)
it's simply represents the intersection points of the two lines
or as i like to describe it as the origin point of the spiral on the complex plane

make sure to check my article that i submitted to Vixra >>> [PDF]

If you are assigning a value for a series that decreases to a specific value (case #1)
Then you can assigning a value for a series that increases from a specific value (case #2) <<< origin point!

Other then those two cases there is one more
This is when the spiral at some point start to spin around a specific value with a “fixed radius”
those cases appears at the zeta function ζ(s)=ζ(x+iy)=a+ib when x=1 and the radius will be 1/y
meaning that this is a divergent series with a “fixed radius”

Its true that the zeta function spirals have 3 cases but they are all spirals with one arm
Now at the eta function the spirals have two arms (that is because of the +/- swapping) with the same 3 cases

By the way the “fixed radius” appears at the eta function η(s)=η(x+iy)=a+ib when Re(s)=0




ζ(s)=ζ(1.1+1.8i) function - spiral with one arm - convergent case , Re(s)>1 ζ(s)=ζ(1+1.8i) function - spiral with one arm - fixed radius case , Re(s)=1 ζ(s)=ζ(0.6+1.8i) function - spiral with one arm - divergent case , Re(s)<1
η(s)=η(0.2+2i) function - spiral with two arms - convergent case , Re(s)>0 η(s)=η(0+2i) function - spiral with two arms - fixed radius case , Re(s)=0 η(s)=η(-0.4+2i) function - spiral with two arms - divergent case , Re(s)<0




Removing the Riemann Hypothesis from the Complex Plane

[Vixra] [PDF] My Email: isaac.mor@hotmail.com

PDF

you dont need the complex plane or the use of analytic continuation in order to see or understand the riemann hypothesis! here is the eta function rotation on the complex plane.

as you can see when ever zeta is getting an "analytic continuation zero" then the eta tunction is also getting a zero but this is a "real zero" because eta is well defined for re(s)>0

same goes when we take eta function summation of the x and y Axis when they are both equal to zero. you dont need the complex plane anymore

Analytic Continuation Zero "Real Zero"
ζ(s)=ζ(0.5+14.1347251417i) function - spiral with one arm - divergent zero η(s)=η(0.5+14.1347251417i) function - spiral with two arms - convergent zero






eta_0.5

















The Riemann Hypothesis Proof

[Vixra] [PDF] My Email: isaac.mor@hotmail.com

PDF





Riemann zeta function formula

[Vixra] [PDF] My Email: isaac.mor@hotmail.com

PDF

Riemann_zeta_function_formula



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