A324997 Decimal expansion of zeta'(-1, 1/6).

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

Offset

0,2

Links

Table of n, a(n) for n=0..105.

J. Miller and V. Adamchik, Derivatives of the Hurwitz Zeta Function for Rational Arguments, Journal of Computational and Applied Mathematics 100 (1998) 201-206. [contains a large number of typos]

Eric Weisstein's World of Mathematics, Hurwitz Zeta Function, formula 25.

Formula

Equals -Pi/(12*sqrt(3)) + log(2)/72 + log(3)/144 + PolyGamma(1, 1/3)/(8*sqrt(3)*Pi) + Zeta'(-1)/6.

A324997 + A324998 = log(2)/36 + log(3)/72 + Zeta'(-1)/3.

Example

0.070452592367204142475462166806035927785155027545830206477019332868362...

Maple

evalf(Zeta(1, -1, 1/6), 120);
evalf(-Pi/(12*sqrt(3)) + log(2)/72 + log(3)/144 + Psi(1, 1/3)/(8*sqrt(3)*Pi) + Zeta(1, -1)/6, 120);

Mathematica

RealDigits[Derivative[1, 0][Zeta][-1, 1/6], 10, 120][[1]]
N[With[{k=1}, -(9^k - 1) * (2^(2*k-1) + 1) * BernoulliB[2*k] * Pi/(8*Sqrt[3]*k*6^(2*k - 1)) + BernoulliB[2*k] * (3^(2*k-1) - 1)*Log[2]/(4*k*6^(2*k - 1)) + BernoulliB[2*k]*(2^(2*k-1) - 1) * Log[3]/(4*k*6^(2*k-1)) - (-1)^k*(2^(2*k-1) + 1) * PolyGamma[2*k-1, 1/3] / (2*Sqrt[3]*(12*Pi)^(2*k - 1))+(2^(2*k - 1) - 1)*(3^(2*k - 1) - 1)*Zeta'[1-2*k]/2/6^(2*k-1)], 120]

Prog

(PARI) zetahurwitz'(-1, 1/6) \\ Michel Marcus, Mar 24 2019

Crossrefs

Cf. A084448, A240966, A324998.

Sequence in context: A065463 A319739 A242780 * A011392 A177156 A016582

Adjacent sequences: A324994 A324995 A324996 * A324998 A324999 A325000

Keyword

nonn,cons

Author

Vaclav Kotesovec, Mar 23 2019

Status

approved