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ID: INPUT{id.yaml}

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Title: >

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

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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

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  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 

    $\sum_{n=0}^\infty n^{-s}, \Re(s) > 1$.

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  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).

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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]}

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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

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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

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Similar tables:

Keywords:

Tags:

- Taylor series

- L-function

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Data properties:

  type: R

  complete: no

Display properties:

  number-header: >

    $n$<sup>th</sup> coefficient

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Numbers: INPUT{numbers.yaml}

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