Dirichlet dynamical zeta function for billiard flow

We study the Dirichlet dynamical zeta function ${\eta }_D(s)$ for billiard flow corresponding to several strictly convex disjoint obstacles. For large $\mathrm{R}\mathrm{e}s$ we have ${\eta }_D(s)={\sum }_{n=1}^{\mathrm{\infty }}{a}_n{e}^{-{\lambda }_{n}s},a_n\in \mathbb{R}$ and ${\eta }_D$ admits a meromorphic continuation to $\mathbb{C}$. We obtain some conditions of the frequencies ${\lambda }_n$ and some sums of coefficients $a_n$ which imply that ${\eta }_D$ cannot be prolonged as entire function.