Given a compact, triangulable space $X$ and a continuous function $f: X\rightarrow X$, then we define the Lefschetz number $\Lambda_{f}$ by $$\Lambda_{f} = \sum_{k\geq0}(-1)^{k}Tr(f_{\ast}\vert H_{k}(X,\mathbb{C}))$$
If $f$ has only finitely many fixed points, then by the Lefschetz-Hopf theorem $$\Lambda_{f}=\sum_{x=f(x)}i(f,x)$$ where $i(f,x)$ is the fixed point degree of the fixed point $x$. The fixed point degree at a point can loosely be thought of as the multiplicity of the map at that point.
We define the Lefschetz Zeta function by $$\zeta_{f}(z)=\exp\left(\sum_{n=1}^{\infty}\Lambda_{f^n}\frac{z^n}{n}\right)$$ and we see using the Lefschetz Hopf theorem that $$\zeta_{f}(z)=\prod_{i=0}^{n}\det(1-zf_{\ast}\vert H_{i}(X,\mathbb{C}))^{(-1)^{i+1}}$$
Now this is all very well and good, but as far as I can tell there is nothing new to be gained from $\zeta_{f}(z)$. It seems to be specifically constructed so that the Lefschetz Hopf theorem can simplify it. Am I wrong? Does $\zeta_{f}(z)$ actually help us retireive informatin about $f$? Do the coefficients of $\zeta_{f}(z)$ correspond to anything? When would I actually learn anything by constructing this function?