Dirichlet Beta Function at Odd Positive Integers

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Theorem

$\displaystyle \map \beta {2 n + 1} = \paren {-1}^n \frac {E_{2 n} \pi^{2 n + 1} } {4^{n + 1} \paren {2 n}!}$

where:

$\beta$ denotes the Dirichlet beta function
$E_n$ denotes the $n$th Euler number
$n$ is a non-negative integer.


Proof