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ID: INPUT{id.yaml}

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

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:
  expression:
    title: Transformation of constant
    type: Symbolic
    show-in-parameter-list: no

n:
    type: Z
    constraints: $n \geq 0$
  
Comments:
  comment-Riemann-zeta-definition: >
    The Riemann zeta function $\zeta(s)$
    is the meromorphic continuation of the Dirichlet series 
    $\sum_{n=0}^\infty n^{-s}, \Re(s) > 1$.

comment-odd-coefficients: >
    $a_n = 0$ for every odd $n$ due to CITE{formula-functional-equation}.
  
Formulas:
  formula-functional-equation: >
    $\xi(s) = \xi(1-s)$ (functional equation).

Programs: 
  program-sage:
    language: Sage
    code: |
      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:
  Kei92:
    bib: >
      J. B. Keiper, "Power series expansions of Riemann's $\xi$ function",
      Math. Comp. 58 (1992), 765-773.
    doi: 10.1090/S0025-5718-1992-1122072-5
    MR: 1122072

Links:
  WikiZeta:
    title: "Wikipedia: Riemann zeta function"
    url: https://en.wikipedia.org/wiki/Riemann_zeta_function
  WikiXi:
    title: "Wikipedia: Riemann Xi function"
    url: https://en.wikipedia.org/wiki/Riemann_Xi_function

Similar tables:
  
Keywords:
  
Tags:
- Taylor series
- L-function

Data properties:
  type: R
  complete: no
  
Display properties:
  number-header: >
    $n$<sup>th</sup> coefficient

Numbers: INPUT{numbers.yaml}

— 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

[2]

Wikipedia: Riemann zeta function

[3]

Wikipedia: Riemann Xi function

Data properties

Entries are of type: real number

Table is complete: no