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A210593 Decimal expansion of the series limit of Sum_{k>=1} (-1)^k*log(k)/k^2.

%I A210593 #34 Aug 18 2024 09:02:43
%S A210593 1,0,1,3,1,6,5,7,8,1,6,3,5,0,4,5,0,1,8,8,6,0,0,2,8,8,2,2,1,2,2,4,2,1,
%T A210593 8,3,6,5,9,3,8,4,7,7,6,3,7,4,9,1,1,1,6,3,3,3,4,2,9,4,2,4,7,1,9,6,2,0,
%U A210593 4,5,3,0,9,2,0,5,4,3,6,3,2,4,9,5,3,1,7,8,0,1,2,5,3,1,9,0,3,5,6,3,9,8,2,3,1
%N A210593 Decimal expansion of the series limit of Sum_{k>=1} (-1)^k*log(k)/k^2.
%C A210593 First derivative of the Dirichlet eta-function eta(s) at s=2.
%C A210593 Phatisena et al. misspell "Euler" and provide the wrong sign and an invalid 7th digit.
%H A210593 G. C. Greubel, <a href="/A210593/b210593.txt">Table of n, a(n) for n = 0..10000</a>
%H A210593 S. Phatisena, R. E. Amritkar, P. V. Panat, <a href="http://dx.doi.org/10.1103/PhysRevA.34.5070">Exchange and correlation potential for a two-dimensional electron gas at finite temperatures</a>, Phys. Rev. A 34 (1986) 5070.
%H A210593 Wikipedia, <a href="http://en.wikipedia.org/wiki/Dirichlet_eta_function">Dirichlet eta function</a>
%F A210593 Decimal expansion of (log(2)*zeta(2) + zeta'(2)) / 2.
%e A210593 0.101316578163504501886002882212242183659384776374911163334294247196204...
%p A210593 1/2*log(2)*Zeta(2)+Zeta(1,2)/2 ; evalf(%) ;
%t A210593 N[(1/12)*Pi^2*(Log[4] - 12*Log[Glaisher] + Log[Pi] + EulerGamma), 105] // RealDigits // First (* _Jean-François Alcover_, Feb 05 2013 *)
%o A210593 (PARI) (log(2)*zeta(2)+zeta'(2))/2 \\ _Charles R Greathouse IV_, Mar 28 2012
%Y A210593 Cf. A073002, A013661, A002162, A091812 (s=1), A375506 (s=3/2), A349220 (s=3), A349252 (s=4).
%K A210593 nonn,cons
%O A210593 0,4
%A A210593 _R. J. Mathar_, Mar 23 2012
%E A210593 Extended to 105 digits by _Jean-François Alcover_, Feb 05 2013