The problem of calculating Hasse--Weil zeta-function for a given curve $C/\mathbb{F}_p$ over a finite field is far from being easy, especially for large genus (as discussed by Wouter Castryck at https://homepages.warwick.ac.uk/~maseap/arith/notes/pointcountingwarwick.pdf).

I'm interested in examples of explicit calculations of Hasse--Weil zeta-functions for curves. Calculations made with the use of heavy machinery, such as $l$-adic cohomology, and calculations made with a more elementary number theory are welcomed.

There is a connection between cocycles classes in the first cohomology group $H^1(C)$ and zeroes of Hasse--Weil zeta. Is there any sources where this connection is illustrated with explicit examples?