A324992 Decimal expansion of zeta'(-1, 1/2).

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

Offset: 0

Author

Vaclav Kotesovec, Mar 23 2019

Keywords

• nonn
• cons

Examples

			0.053829439326894410047908491727299631045539017902590256244899486116455...
		

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 22.

Crossrefs

Cf. A074962, A084448, A240966, A261829, A306635.

Programs

Maple

evalf(Zeta(1,-1,1/2), 120);
evalf(-log(2)/24 - Zeta(1,-1)/2, 120);

Mathematica

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

PARI

zetahurwitz'(-1, 1/2) \\ Michel Marcus, Mar 24 2019

Formula

Equals -log(2)/24 - Zeta'(-1)/2 = A261829 - log(2)/24.

Equals -1/24 - log(2)/24 + log(A)/2, where A is the Glaisher-Kinkelin constant A074962.

Equals (log(Pi) - 1 + gamma)/24 - Zeta'(2)/(4*Pi^2), where gamma is the Euler-Mascheroni constant A001620.