A324996 Decimal expansion of zeta'(-1, 3/4) (negated).

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

Offset: 0

Author

Vaclav Kotesovec, Mar 23 2019

Keywords

• nonn
• cons

Examples

			-0.05221258304514832888285615658675114937051199095455909460693510398326...
		

Links

• 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.

Crossrefs

Cf. A084448, A240966, A324995.

Programs

Maple

evalf(Zeta(1,-1,3/4), 120);
evalf(Pi/32 - Psi(1, 1/4)/(32*Pi) - Zeta(1,-1)/8, 120);

Mathematica

RealDigits[Derivative[1, 0][Zeta][-1, 3/4], 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]

PARI

zetahurwitz'(-1, 3/4) \\ Michel Marcus, Mar 24 2019

Formula

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

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