We consider a topological dynamical system $(\Sigma , \sigma ) on the real line which is topologically conjugate to a topologically mixing one-sided subshift <formula form="inline" disc="math" id="ffm002"><formtex notation="AMSTeX">$(\Sigma_{A}^{+},\sigma_A )$ of finite type with structure matrix $A$. Moreover, we assume that it satisfies some additional conditions so that the map $\sigma$ can have an expanding extension with nice properties in a neighborhood of $\Sigma$. Let $u$ be an eventually positive Lipschitz continuous function on $\Sigma$ and $\zeta_u (s)$ the corresponding dynamical zeta function. We show that $\zeta_u (s)$ has a meromorphic extension in the half-plane $\text{Re }s > -\beta_u$ for some $\beta_u >0$ and its special value at the origin is given by $\zeta_u (0) =1/\text{det}(I-A)$. As an application, we can see that the zeta function $\zeta_Q (s)$ for the two-dimensional dispersing billiard table $Q$ without eclipse has a meromorphic extension in the half-plane $\text{Re }s > -\beta$ for some $\beta>0$ and $\zeta_Q (0)=-1/(J-2)2^{J-1}$ holds, where $J$ is the number of scatterers.