  Let $K$ be the function field of a curve over a finite field of odd
characteristic. We investigate using $L$-functions of Galois extensions of $K$
to effectively recover $K$. When $K$ is the function field of the projective
line with four rational points removed, we show how to use $L$-functions of a
ray class field to effectively recover the removed points up to automorphisms
of the projective line. When $K$ is the function field of a plane curve, we
show how to effectively recover the equation of that curve using $L$-functions
of Artin-Schreier extensions of $K$.
