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ID: INPUT{id.yaml}

Title: >
  Zeros of the Riemann zeta function

Definition: >
  The list contains the imaginary parts of the non-trivial zeros of the 
  Riemann zeta function, ordered increasingly.

Parameters:
  n:
    type: Z
    constraints: $n \neq 0$
    comments: >
      The positive imaginary parts are labelled $n = 1, 2, 3, \ldots$, 
      the negative ones are labelled $n = -1, -2, -3, \ldots$,
      and the index $n = 0$ is skipped.
   
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-non-trivial-zeros: >
    Its so-called trivial zeros are the negative even integers, 
    which are therefore not listed.

All known non-trivial zeros lie on the line $\Re(s) = 1/2$.
    Their imaginary parts are symmetric about 0; 
    whereby it is enough to know the positive ones.
  
Formulas:

Programs:

References:
  Pla15: 
    bib: David J. Platt, Math. Comp. 84 (2015), 1521-1535.
    doi: 10.1090/S0025-5718-2014-02884-6 
    MR: 3315519

Links:
  Wiki:
    title: "Wikipedia: Riemann zeta function"
    url: https://en.wikipedia.org/wiki/Riemann_zeta_function
  LMFDB:
    title: "LMFDB: Zeta zeros"
    url: https://www.lmfdb.org/zeros/zeta/

Similar tables:
  
Keywords:
- RH
- GRH
  
Tags:
- zero
- L-function

Data properties:
  type: R
  complete: no
  #absolute precision: 2.5e-31
  both signs: True
  sources:
  - CITE{Pla15}
  - CITE{LMFDB}

Display properties:
  number-header: $\rho_n$

Numbers: INPUT{numbers.yaml}

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Zeros of the Riemann zeta function

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Numbers

$n$

$\rho_n$

INPUT{numbers.yaml} (not shown in preview)

Definition

The list contains the imaginary parts of the non-trivial zeros of the Riemann zeta function, ordered increasingly.

Parameters

$n$

— integer ($n \neq 0$)

Comments

(1)

The Riemann zeta function $\zeta(s)$ is the meromorphic continuation of the Dirichlet series $\sum_{n=0}^\infty n^{-s}, \Re(s) > 1$.

(2)

Its so-called trivial zeros are the negative even integers, which are therefore not listed.
All known non-trivial zeros lie on the line $\Re(s) = 1/2$. Their imaginary parts are symmetric about 0; whereby it is enough to know the positive ones.

References

[1]

David J. Platt, Math. Comp. 84 (2015), 1521-1535. (doi)

Links

[2]

Wikipedia: Riemann zeta function

[3]

LMFDB: Zeta zeros

Data properties

Entries are of type: real number

Table is complete: no

both signs: True (Unknown key)

Sources of data: [1], [3]