%I #31 Nov 18 2019 01:35:28

%S 5,7,5,5,4,2,7,1,4,4,6,1,1,7,7,4,5,2,4,3,1,1,0,6,4,0,5,4,9,2,8,6,3,8,

%T 3,3,5,6,7,4,5,6,6,1,5,1,7,9,7,9,9,5,3,9,5,2,9,2,4,7,5,8,1,9,3,5,9,5,

%U 4,5,2,1,3,8,3,6,2,3,6,4,0,7,8,1,9,0,1,6,3,1,0,0,5,4,8,5,8,9,4,7,2,3

%N Decimal expansion of lambda(5) 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 lambda(5) = 5*Pi^2/4 + 5*Pi^4/96 - 5*log(4)/2 - 5*log(Pi)/2 + 5*gamma/2 - 10*gamma^2 + 10*gamma^3 - 5*gamma^4+gamma^5 - 20*gamma(1) + 30*gamma*gamma(1) - 20*gamma^2*gamma(1) + 5*gamma^3*gamma(1) - 10*gamma(1)^2 + 5*gamma*gamma(1)^2 + 15*gamma(2) - 10*gamma*gamma(2) + 5/2*gamma^2*gamma(2) + 5/2*gamma(1)*gamma(2) - 10*gamma(3)/3 + 5/6*gamma*gamma(3) + 5*gamma(4)/24 - 35*zeta(3)/4 - 31*zeta(5)/32+1. - _Jean-François Alcover_, Jul 02 2014

%e 0.575542714...

%t lambda[n_] := Limit[D[s^(n - 1)*Log[xi[s]], {s, n}], s -> 1]/(n - 1)!; RealDigits[N[lambda[5], 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 + 5 e/2 - 10 e^2 + 10 e^3 - 5 e^4 + e^5 + 5 Pi^2/4 + (5 Pi^4)/96 - 20 g[1] + 30 e g[1] - 20 e^2 g[1] + 5 e^3 g[1] - 10 g[1]^2 + 5 e g[1]^2 + 15 g[2] - 10 e g[2] + 5/2 e^2 g[2] + 5/2 g[1] g[2] - 10 g[3]/3 + 5/6 e g[3] + 5 g[4]/24 - Log[32] - 5 Log[Pi]/2 - 35 Zeta[3]/4 - 31 Zeta[5]/32], 10, 110][[1]] (* _Eric W. Weisstein_, Feb 08 2019 *)

%Y Cf. A074760 (lambda_1), A104539 (lambda_2), A104540 (lambda_3), A104541 (lambda_4).

%Y Cf. A306339 (lambda_6), A306340 (lambda_7), A306341 (lambda_8).

%K nonn,cons

%O 0,1

%A _Eric W. Weisstein_, Mar 13 2005