In general, is there a simple way of computing the zeta function of a curve (or variety) over $\mathbb{F}_q$? Here $q$ is an odd prime power. I've seen a nice computation for both affine and projective n-spaces over $\mathbb{F}_q$.
Also, why are we interested in the zeta function of a curve (of variety)? What does it actually tell us about the original object?