A104540 Decimal expansion of lambda(3) in Li's criterion.

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

Offset

0,1

References

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.32, p. 204.

Links

Table of n, a(n) for n=0..101.

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(3) = 3*Pi^2/8 - 3*log(2) - 3*log(Pi)/2 + 3*gamma/2 - 3*gamma^2 + gamma^3 + 3*gamma*gamma(1) - 6*gamma(1) + 3*gamma(2)/2 - 7*zeta(3)/8 + 1. - Jean-François Alcover, Jul 02 2014

Example

0.207638920...

Mathematica

lambda[n_] := Limit[D[s^(n - 1) Log[RiemannXi[s]], {s, n}], s -> 1]/(n - 1)!; RealDigits[N[lambda[3], 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 + 3 e/2 - 3 e^2 + e^3 + 3 Pi^2/8 - 6 g[1] + 3 e g[1] + 3 g[2]/2 - Log[8] - 3 Log[Pi]/2 - 7 Zeta[3]/8], 10, 110][[1]] (* Eric W. Weisstein, Feb 08 2019 *)

Crossrefs

Cf. A074760 (lambda_1), A104539 (lambda_2), A104541 (lambda_4), A104542 (lambda_5).

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

Sequence in context: A351727 A337450 A228819 * A178818 A354615 A113319

Adjacent sequences: A104537 A104538 A104539 * A104541 A104542 A104543

Keyword

nonn,cons

Author

Eric W. Weisstein, Mar 13 2005

Status

approved