  We prove that at least one of the six numbers $\beta(2i)$ for $i=1,\dots,6$
is irrational. Here $\beta(s)=\sum_{k=0}^\infty(-1)^k(2k+1)^{-s}$ denotes
Dirichlet's beta function, so that $\beta(2)$ is Catalan's constant.
