\begin{definition}[Definition:Dirichlet Beta Function]
The '''Dirichlet beta function''' $\beta$ is the complex function defined on the half-plane $\map \Re s > 0$ by the series: 
:$\ds \map \beta s = \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n} {\paren {2 n + 1}^s}$
\end{definition}