A271533 Decimal expansion of the derivative of the Dirichlet function eta(z) at z = -1.

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

Offset: 0

Author

Stanislav Sykora, Apr 09 2016

Keywords

• nonn
• cons

Comments

This entry completes the values of the derivatives eta'(z) at z = 0,1,i,-1,-i (see crossrefs).

Examples

			0.265214370914704351169348273575616405600275762885520266292673582574...
		

Links

• Stanislav Sykora, Table of n, a(n) for n = 0..2000
• Eric Weisstein's World of Mathematics, Dirichlet Eta Function

Crossrefs

Cf. A074962, A256358 (eta'(0)), A091812 (eta'(1)), A271525 (real(eta'(i))), A271526 (-imag(eta'(i))) .

Programs

Mathematica

RealDigits[3*Log[Glaisher] - Log[2]/3 - 1/4, 10, 120][[1]] (* G. C. Greubel, Apr 09 2016 *)
RealDigits[DirichletEta'[-1], 10, 110][[1]] (* Eric W. Weisstein, Jan 06 2024 *)

PARI

\\ Derivative of Dirichlet eta function (fails for z=1):
derdireta(z)=2^(1-z)*log(2)*zeta(z)+(1-2^(1-z))*zeta'(z);
derdireta(-1) \\ Evaluation

Formula

eta'(-1) = 3*log(A) - log(2)/3 - 1/4, where A = A074962 is the Glaisher-Kinkelin constant.