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 A104542 Decimal expansion of lambda(5) in Li's criterion. 15
 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, 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, 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 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 LINKS E. Bombieri and J. C. Lagarias, Complements to Li's Criterion for the Riemann Hypothesis, J. Number Th. 77(2) (1999), 274-287. M. W. Coffey, Relations and positivity results for derivatives of the Riemann xi function, J. Comput. Appl. Math. 166(2) (2004), 525-534. Xian-Jin Li, The positivity of a sequence of numbers and the Riemann hypothesis, J. Number Th. 65(2) (1997), 325-333. Eric Weisstein's World of Mathematics, Li's Criterion. Eric Weisstein's World of Mathematics, Riemann Zeta Function Zeros. Wikipedia, Li's criterion. FORMULA 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 EXAMPLE 0.575542714... MATHEMATICA 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 *) 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 *) CROSSREFS Cf. A074760 (lambda_1), A104539 (lambda_2), A104540 (lambda_3), A104541 (lambda_4). Cf. A306339 (lambda_6), A306340 (lambda_7), A306341 (lambda_8). Sequence in context: A198744 A201944 A165242 * A161376 A107437 A317083 Adjacent sequences:  A104539 A104540 A104541 * A104543 A104544 A104545 KEYWORD nonn,cons AUTHOR Eric W. Weisstein, Mar 13 2005 STATUS approved

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Last modified September 17 01:53 EDT 2021. Contains 347478 sequences. (Running on oeis4.)