  In a recent work, Dancs and He found an Euler-type formula for
$\,\zeta{(2\,n+1)}$, $\,n\,$ being a positive integer, which contains a series
they could not reduce to a finite closed-form. This open problem reveals a
greater complexity in comparison to $\zeta(2n)$, which is a rational multiple
of $\pi^{2n}$. For the Dirichlet beta function, the things are `inverse':
$\beta(2n+1)$ is a rational multiple of $\pi^{2n+1}$ and no closed-form
expression is known for $\beta(2n)$. Here in this work, I modify the Dancs-He
approach in order to derive an Euler-type formula for $\,\beta{(2n)}$,
including $\,\beta{(2)} = G$, the Catalan's constant. I also convert the
resulting series into zeta series, which yields new exact closed-form
expressions for a class of zeta series involving $\,\beta{(2n)}$ and a finite
number of odd zeta values. A closed-form expression for a certain zeta series
is also conjectured.
