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

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, 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, 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

Offset: 0

Author

R. J. Mathar, Mar 23 2012

Keywords

• nonn
• cons

Comments

First derivative of the Dirichlet eta-function eta(s) at s=2.

Phatisena et al. misspell "Euler" and provide the wrong sign and an invalid 7th digit.

Examples

			0.101316578163504501886002882212242183659384776374911163334294247196204...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• S. Phatisena, R. E. Amritkar, P. V. Panat, Exchange and correlation potential for a two-dimensional electron gas at finite temperatures, Phys. Rev. A 34 (1986) 5070.
• Wikipedia, Dirichlet eta function

Crossrefs

Cf. A073002, A013661, A002162, A091812 (s=1), A375506 (s=3/2), A349220 (s=3), A349252 (s=4).

Programs

Maple

1/2*log(2)*Zeta(2)+Zeta(1,2)/2 ; evalf(%) ;

Mathematica

N[(1/12)*Pi^2*(Log[4] - 12*Log[Glaisher] + Log[Pi] + EulerGamma), 105] // RealDigits // First (* Jean-François Alcover, Feb 05 2013 *)

PARI

(log(2)*zeta(2)+zeta'(2))/2 \\ Charles R Greathouse IV, Mar 28 2012

Formula

Decimal expansion of (log(2)*zeta(2) + zeta'(2)) / 2.

Extensions

Extended to 105 digits by Jean-François Alcover, Feb 05 2013