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A104542
Decimal expansion of lambda(5) in Li's criterion.
18
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
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: A201944 A377932 A165242 * A161376 A107437 A317083
KEYWORD
nonn,cons
AUTHOR
Eric W. Weisstein, Mar 13 2005
STATUS
approved