Recurrence Relations for $\beta (2k)$ and $\zeta (2k+1)$

In this work we study integrals of the form ${\int }_0^{\mathrm{\infty }}\frac{\tanh (x)}{x}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}(x{)}^L\exp (-Tx)dx$. For $L\in \mathbb{N}$, $L\le 4$ and $T\in \mathbb{R}$ we give explicit expressions in terms of derivatives of the Hurwitz zeta function at negative integers. We use these expressions to evaluate these integrals for $T\in {\mathbb{N}}_0$ exactly. For the special case $T=0$ we give explicit evaluations for any $L\in \mathbb{N}$ based on the functional equations for $\beta (s)$ and $\zeta (s)$. As it turns out the value of ${\int }_0^{\mathrm{\infty }}\frac{\tanh (x)}{x}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}(x{)}^{2N+1}dx$ is a linear combination of $\beta (2)$, …, $\beta (2N+2)$ and the value of ${\int }_0^{\mathrm{\infty }}\frac{\tanh (x)}{x}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}(x{)}^{2N}dx$ a linear combination of $\zeta (3)$, …, $\zeta (2N+1)$. We give recursive formulae for the coefficients in these linear combinations.