A271533 - OEIS

%I #17 Jan 06 2024 16:19:02

%S 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,

%T 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,

%U 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

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

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

%H Stanislav Sykora, <a href="/A271533/b271533.txt">Table of n, a(n) for n = 0..2000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DirichletEtaFunction.html">Dirichlet Eta Function</a>

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

%e 0.265214370914704351169348273575616405600275762885520266292673582574...

%t RealDigits[3*Log[Glaisher] - Log[2]/3 - 1/4, 10, 120][[1]] (* _G. C. Greubel_, Apr 09 2016 *)

%t RealDigits[DirichletEta'[-1], 10, 110][[1]] (* _Eric W. Weisstein_, Jan 06 2024 *)

%o (PARI) \\ Derivative of Dirichlet eta function (fails for z=1):

%o derdireta(z)=2^(1-z)*log(2)*zeta(z)+(1-2^(1-z))*zeta'(z);

%o derdireta(-1) \\ Evaluation

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

%K nonn,cons

%O 0,1

%A _Stanislav Sykora_, Apr 09 2016