A324995 Decimal expansion of zeta'(-1, 1/4).

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

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 24.

Formula

Equals -Pi/32 + PolyGamma(1, 1/4)/(32*Pi) - Zeta'(-1)/8.

A324995 + A324996 = -Zeta'(-1)/4.

Equals A006752/(4*Pi) + log(A074962)/8 - 1/96. - Artur Jasinski, Feb 23 2023

Example

0.093567868970261061186336071647446310061521086038359540523565680572606...

Maple

evalf(Zeta(1, -1, 1/4), 120);
evalf(-Pi/32 + Psi(1, 1/4)/(32*Pi) - Zeta(1, -1)/8, 120);

Mathematica

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

Prog

(PARI) zetahurwitz'(-1, 1/4) \\ Michel Marcus, Mar 24 2019

Crossrefs

Cf. A084448, A240966, A324996.

Sequence in context: A371857 A371604 A145924 * A200007 A153618 A171051

Adjacent sequences: A324992 A324993 A324994 * A324996 A324997 A324998

Keyword

nonn,cons

Author

Vaclav Kotesovec, Mar 23 2019

Status

approved