cp's OEIS Frontend

A306341 Decimal expansion of lambda(8) in Li's criterion.

%I A306341 #15 Feb 16 2025 08:33:55
%S A306341 1,4,6,5,7,5,5,6,7,7,1,4,7,0,6,0,6,3,2,6,5,5,5,1,4,5,4,1,9,7,7,7,4,8,
%T A306341 7,8,7,9,1,9,8,4,7,8,6,1,8,7,4,5,4,4,4,6,5,8,4,5,8,5,7,7,5,3,8,3,5,7,
%U A306341 9,5,0,2,8,5,2,3,5,6,3,7,9,4,4,7,8,1,1,5,1,7,5,5,6,0,3,8,0,6,9,0,0,1,2,5,0,1,5,1,8,6
%N A306341 Decimal expansion of lambda(8) in Li's criterion.
%H A306341 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 A306341 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 A306341 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 A306341 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LisCriterion.html">Li's Criterion</a>.
%H A306341 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RiemannZetaFunctionZeros.html">Riemann Zeta Function Zeros</a>.
%H A306341 Wikipedia, <a href="https://en.wikipedia.org/wiki/Li%27s_criterion">Li's criterion</a>.
%e A306341 1.465755677147060632...
%t A306341 RealDigits[With[{e = EulerGamma, g = StieltjesGamma}, 1 + 4 e - 28 e^2 + 56 e^3 - 70 e^4 + 56 e^5 - 28 e^6 + 8 e^7 - e^8 + 7 Pi^2/2 + 35 Pi^4/48 + 7 Pi^6/240 + 17 Pi^8/161280 - 56 g[1] + 168 e g[1] - 280 e^2 g[1] + 280 e^3 g[1] - 168 e^4 g[1] + 56 e^5 g[1] - 8 e^6 g[1] - 140 g[1]^2 + 280 e g[1]^2 - 252 e^2 g[1]^2 + 112 e^3 g[1]^2 - 20 e^4 g[1]^2 - 56 g[1]^3 + 56 e g[1]^3 - 16 e^2 g[1]^3 - 2 g[1]^4 + 84 g[2] - 140 e g[2] + 140 e^2 g[2] - 84 e^3 g[2] + 28 e^4 g[2] - 4 e^5 g[2] + 140 g[1] g[2] - 168 e g[1] g[2] + 84 e^2 g[1] g[2] - 16 e^3 g[1] g[2] + 28 g[1]^2 g[2] - 12 e g[1]^2 g[2] - 21 g[2]^2 + 14 e g[2]^2 - 3 e^2 g[2]^2 - 2 g[1] g[2]^2 - 140 g[3]/3 + 140/3 e g[3] - 28 e^2 g[3] + 28/3 e^3 g[3] - 4/3 e^4 g[3] - 28 g[1] g[3] + 56/3 e g[1] g[3] - 4 e^2 g[1] g[3] - 4/3 g[1]^2 g[3] + 14/3 g[2] g[3] - 4/3 e g[2] g[3] - g[3]^2/9 + 35 g[4]/3 - 7 e g[4] + 7/3 e^2 g[4] - 1/3 e^3 g[4] + 7/3 g[1] g[4] - 2/3 e g[1] g[4] - 1/6 g[2] g[4] - 7 g[5]/5 + 7/15 e g[5] - 1/15 e^2 g[5] - 1/15 g[1] g[5] + 7 g[6]/90 - 1/90 e g[6] - g[7]/630 - 4 Log[4 Pi] - 49 Zeta[3] - 217 Zeta[5]/4 - 127 Zeta[7]/16], 10, 110][[1]]
%Y A306341 Cf. A074760 (lambda_1), A104539 (lambda_2), A104540 (lambda_3), A104541 (lambda_4).
%Y A306341 Cf. A104542 (lambda_5), A306339 (lambda_6), A306340 (lambda_7).
%K A306341 nonn,cons
%O A306341 1,2
%A A306341 _Eric W. Weisstein_, Feb 08 2019