Dirichlet Beta Function at Odd Positive Integers

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.