— preview —

Taylor coefficients of the completed Riemann zeta function at 1/2

edit on github
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).

Comments

(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)

Links

Data properties

Entries are of type: real number

Table is complete: no