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Taylor coefficients of the completed Riemann zeta function at 1/2
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Numbers
$n$
$n$th coefficient
INPUT{numbers.yaml} (not shown in preview)
Definition
The list contains the coefficients of the Taylor expansion $\xi(s) = \sum_{n=0}^\infty \frac{a_n}{n!} (s-1/2)^n$ of the completed Riemann zeta function $\xi(s) = \frac{1}{2}s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)$ at $s=1/2$.
Parameters
$n$
—   integer ($n \geq 0$)
Formulas
(1)
$\xi(s) = \xi(1-s)$ (functional equation).
(2)
The Riemann zeta function $\zeta(s)$ is the meromorphic continuation of the Dirichlet series $\sum_{n=0}^\infty n^{-s}, \Re(s) > 1$.
(3)
$a_n = 0$ for every odd $n$ due to (1).
Programs
(P1)
Sage
s = var("s")
xi(s) = 1/2 * s*(s-1)*pi^(-s/2)*gamma(s/2)*zeta(s)
numbers = {n: xi.derivative(s,n)(s=1/2) for n in [0..10]}

References
[1]
J. B. Keiper, "Power series expansions of Riemann's $\xi$ function", Math. Comp. 58 (1992), 765-773. (doi)