A013670 - OEIS

A013670

Decimal expansion of zeta(12).

1, 0, 0, 0, 2, 4, 6, 0, 8, 6, 5, 5, 3, 3, 0, 8, 0, 4, 8, 2, 9, 8, 6, 3, 7, 9, 9, 8, 0, 4, 7, 7, 3, 9, 6, 7, 0, 9, 6, 0, 4, 1, 6, 0, 8, 8, 4, 5, 8, 0, 0, 3, 4, 0, 4, 5, 3, 3, 0, 4, 0, 9, 5, 2, 1, 3, 3, 2, 5, 2, 0, 1, 9, 6, 8, 1, 9, 4, 0, 9, 1, 3, 0, 4, 9, 0, 4, 2, 8, 0, 8, 5, 5, 1, 9, 0, 0, 6, 9

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OFFSET

1,5

REFERENCES

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 811.

LINKS

Table of n, a(n) for n=1..99.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

FORMULA

zeta(12) = 2/3*2^12/(2^12 - 1)*( Sum_{n even} n^2*p(n)/(n^2 - 1)^13 ), where p(n) = 7*n^12 + 182*n^10 + 1001*n^8 + 1716*n^6 + 1001*n^4 + 182*n^2 + 7 is a row polynomial of A091043. - Peter Bala, Dec 05 2013

zeta(12) = Sum_{n >= 1} (A010052(n)/n^6) = Sum {n >= 1} ( (floor(sqrt(n))-floor(sqrt(n-1)))/n^6 ). - Mikael Aaltonen, Feb 20 2015

zeta(12) = (691/638512875)*Pi^12 (see A046988/A002432). - Rick L. Shepherd, May 30 2016

zeta(12) = Product_{k>=1} 1/(1 - 1/prime(k)^12). - Vaclav Kotesovec, May 02 2020

zeta(12) = Sum_{k=1..(n-1)} (Gamma(k/n)*Gamma(1 - k/n))^12 / (2*(n^12 + (13650/691)*n^10 + (279279/1382)*n^8 + (993850/691)*n^6 + (5753748/691)*n^4 + (39312000/691)*n^2 - 92427157/1382)). - Andrea Pinos, May 05 2026

EXAMPLE

1.0002460865533080482986379980477396709604160884580034045330409521332520...

MATHEMATICA

RealDigits[Zeta[12], 10, 120][[1]] (* Harvey P. Dale, Apr 30 2013 *)

PROG

(PARI) zeta(12) \\ Michel Marcus, Feb 20 2015

CROSSREFS

Cf. A013662, A013664, A013666, A013668.

Cf. A002432, A046988.

Cf. A010052, A091043.

Sequence in context: A141062 A131806 A004518 * A121206 A062004 A300268

Adjacent sequences: A013667 A013668 A013669 * A013671 A013672 A013673

KEYWORD

cons,nonn,changed

AUTHOR

N. J. A. Sloane

STATUS

approved