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%I #10 Mar 07 2023 10:31:34
%S 6,7,0,9,0,8,3,0,7,8,8,2,4,7,8,8,0,6,0,8,5,2,7,1,5,9,9,2,5,3,8,5,3,4,
%T 2,6,8,1,6,2,6,0,9,7,1,7,9,7,6,7,2,5,3,5,0,5,8,3,6,1,7,6,7,5,0,0,0,7,
%U 0,3,2,9,9,9,4,3,6,8,4,9,8,6,2,5,8,2,4,1,4,7,5,3,0,8,5,9,6,1,9,4,5,5,4
%N Decimal expansion of gamma_2, a lattice sum constant, analog of Euler's constant for two-dimensional lattices.
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Sections 1.10 Madelung's constant, p. 80.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LatticeSum.html">Lattice Sum</a>.
%F gamma_2 = (1/4)*(delta_2 + 2*log((sqrt(2) + 1)/(sqrt(2) - 1)) - 4*EulerGamma), where delta_2 is A247042.
%e -0.670908307882478806085271599253853426816260971797672535...
%t delta2 = 2*Zeta[1/2]*(Zeta[1/2, 1/4] - Zeta[1/2, 3/4]); gamma2 = (1/4)*(delta2 + 2*Log[(Sqrt[2] + 1)/(Sqrt[2] - 1)] - 4*EulerGamma); RealDigits[gamma2, 10, 103] // First
%o (PARI) (2*zeta(1/2)*(zetahurwitz(1/2, 1/4)-zetahurwitz(1/2, 3/4)) + 2*log((sqrt(2) + 1)/(sqrt(2) - 1)))/4 - Euler \\ _Charles R Greathouse IV_, Jan 31 2018
%Y Cf. A247042.
%K nonn,cons,easy
%O 0,1
%A _Jean-François Alcover_, Sep 10 2014
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