Dirichlet beta
From specialfunctionswiki
The Dirichlet $\beta$ function is defined by $$\beta(x) = \displaystyle\sum_{k=0}^{\infty} (-1)^k (2k+1)^{-x}.$$
Graph of $\beta$ on $[-4,4]$.
Domain coloring of analytic continuation of $\beta$.
Properties
Catalan's constant using Dirichlet beta
Dirichlet beta in terms of Lerch transcendent
