  It is commonly known that $\zeta(2k) = q_{k}\frac{\zeta(2k + 2)}{\pi^2}$ with
known rational numbers $q_{k}$. In this work we construct recurrence relations
of the form $\sum_{k = 1}^{\infty}r_{k}\frac{\zeta(2k + 1)}{\pi^{2k}} = 0$ and
show that series representations for the coefficients $r_{k} \in \mathbb{R}$
can be computed explicitly.
