Dirichlet beta function

The Dirichlet beta function is

β (s) = \sum_{n \geq 0} (- 1)^{n} (2 n + 1)^{- s}

and is the Dirichlet L-function for the nontrivial character modulo four.

Table of contents
- Special values of the Dirichlet beta function Dirichlet beta function
- Dirichlet beta reflection formula Dirichlet beta function

Special values of the Dirichlet beta function

Hankel residue extraction gives

β (0) = 1/2

,

β (- 2) = - 1/2

, and values at other nonpositive integers from the Taylor coefficients of

1/ (2 cosh t)

.

Dirichlet beta reflection formula

The functional equation may be written

β (1 - s) = Γ (s) (π /2)^{- s} sin (π s /2) β (s) .

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Trivial zero of a Dirichlet L-function

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Dirichlet beta function by

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The Dirichlet beta function, denoted as \( \beta(s) \), is a special function that generalizes the concept of the Riemann zeta function.

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