In this note, we study the $q$-Dirichlet series $Z_q(s)$ which was evaluated at non-positive integers by Satoh. We consider the values of $Z_q(s)$ at positive integers. By letting $q \to 1$, we get the Euler formulas for $\zeta(2k)$ and the recent formulas for $\zeta(2k+1)$ given by Cvijović-Klinowski. We also consider the relation between $Z_q(s)$ and Jackson's $q$-$\Gamma$-function.
Hirofumi Tsumura. "A note on $q$-analogues of Dirichlet series." Proc. Japan Acad. Ser. A Math. Sci. 75 (3) 23 - 25, March 1999. https://doi.org/10.3792/pjaa.75.23