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
3*Pi^2/4 + Pi^4/96 - 2*log(4) - 2*log(Pi) + 2*gamma - 6*gamma^2 + 4*gamma^3 - gamma^4 - 12*gamma(1) + 12*gamma*gamma(1) - 4*gamma^2*gamma(1) - 2*gamma(1)^2 + 6*gamma(2) - 2*gamma*gamma(2) - 2*gamma(3)/3 - 7*zeta(3)/2 + 1. - Jean-François Alcover, Jul 02 2014
EXAMPLE
0.368790479...
MATHEMATICA
lambda[n_] := Limit[D[s^(n - 1) Log[RiemannXi[s]], {s, n}], s -> 1]/(n - 1)!; RealDigits[N[lambda[4], 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 + 2 e - 6 e^2 + 4 e^3 - e^4 + 3 Pi^2/4 + Pi^4/96 - 12 g[1] + 12 e g[1] - 4 e^2 g[1] - 2 g[1]^2 + 6 g[2] - 2 e g[2] - 2 g[3]/3 - 2 Log[4 Pi] - 7 Zeta[3]/2], 10, 110][[1]] (* Eric W. Weisstein, Feb 08 2019 *)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Eric W. Weisstein, Mar 13 2005
STATUS
approved