The behavior of the dynamical zeta function ${{Z}_{D}}(s)$ related to several strictly convex disjoint obstacles is similar to that of the inverse $Q(s)\,=\,\frac{1}{\zeta (s)}$ of the Riemann zeta function $\zeta \left( s \right)$. Let $\prod \left( s \right)$ be the series obtained from ${{Z}_{D}}(s)$ summing only over primitive periodic rays. In this paper we examine the analytic singularities of ${{Z}_{D}}(s)$ and $\prod \left( s \right)$ close to the line $\Re s={{s}_{2}},$ where ${{s}_{2}}$ is the abscissa of absolute convergence of the series obtained by the second iterations of the primitive periodic rays. We show that at least one of the functions ${{Z}_{D}}(s),$$\prod \left( s \right)$ has a singularity at $s\,=\,{{s}_{2}}$.