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%I #31 Nov 18 2019 01:35:41
%S 3,6,8,7,9,0,4,7,9,4,9,2,2,4,1,6,3,8,5,9,0,5,1,1,4,8,9,6,3,7,7,5,6,0,
%T 7,2,2,6,2,1,6,6,6,9,3,9,6,0,8,5,2,8,0,4,8,2,3,1,1,8,8,5,6,8,5,0,9,4,
%U 6,2,5,3,2,2,6,5,7,7,9,0,2,6,2,9,0,3,1,5,2,8,3,9,8,6,0,1,5,5,8,4,2,1
%N Decimal expansion of lambda(4) in Li's criterion.
%H E. Bombieri and J. C. Lagarias, <a href="https://doi.org/10.1006/jnth.1999.2392">Complements to Li's Criterion for the Riemann Hypothesis</a>, J. Number Th. 77(2) (1999), 274-287.
%H M. W. Coffey, <a href="https://doi.org/10.1016/j.cam.2003.09.003">Relations and positivity results for derivatives of the Riemann xi function</a>, J. Comput. Appl. Math. 166(2) (2004), 525-534.
%H Xian-Jin Li, <a href="https://doi.org/10.1006/jnth.1997.2137">The positivity of a sequence of numbers and the Riemann hypothesis</a>, J. Number Th. 65(2) (1997), 325-333.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LisCriterion.html">Li's Criterion</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RiemannZetaFunctionZeros.html">Riemann Zeta Function Zeros</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Li%27s_criterion">Li's criterion</a>.
%F 3*Pi^2/4 + Pi^4/96 - 2*log(4) - 2*log(Pi) + 2*gamma - 6*gamma^2 + 4*gamma^3 - gamma^4 - 12*gamma(1) + 12*gamma*gamma(1) - 4*gamma^2*gamma(1) - 2*gamma(1)^2 + 6*gamma(2) - 2*gamma*gamma(2) - 2*gamma(3)/3 - 7*zeta(3)/2 + 1. - _Jean-François Alcover_, Jul 02 2014
%e 0.368790479...
%t lambda[n_] := Limit[D[s^(n - 1) Log[RiemannXi[s]], {s, n}], s -> 1]/(n - 1)!; RealDigits[N[lambda[4], 110]][[1]][[1 ;; 102]] (* _Jean-François Alcover_, Oct 31 2012, after _Eric W. Weisstein_, updated May 18 2016 *)
%t RealDigits[With[{e = EulerGamma, g = StieltjesGamma}, 1 + 2 e - 6 e^2 + 4 e^3 - e^4 + 3 Pi^2/4 + Pi^4/96 - 12 g[1] + 12 e g[1] - 4 e^2 g[1] - 2 g[1]^2 + 6 g[2] - 2 e g[2] - 2 g[3]/3 - 2 Log[4 Pi] - 7 Zeta[3]/2], 10, 110][[1]] (* _Eric W. Weisstein_, Feb 08 2019 *)
%Y Cf. A074760 (lambda_1), A104539 (lambda_2), A104540 (lambda_3), A104542 (lambda_5).
%Y Cf. A306339 (lambda_6), A306340 (lambda_7), A306341 (lambda_8).
%K nonn,cons
%O 0,1
%A _Eric W. Weisstein_, Mar 13 2005
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