Let \(E\) be an elliptic curve with non-constant \(j\)-invariant over a function field \(K\) with constant field of size an odd prime power \(q\). Its \(L\)-function \(L(T,E/K)\) belongs to \(1 + T\mathbb{Z}[T]\). Inspired by the algorithms of Schoof and Pila for computing zeta functions of curves over finite fields, we propose an approach to compute \(L(T,E/K)\). The idea is to compute, for sufficiently many primes \(\ell\) invertible in \(K\), the reduction \(L(T,E/K) \bmod{\ell}\). The \(L\)-function is then recovered via the Chinese remainder theorem. When \(E(K)\) has a subgroup of order \(N \geq 2\) coprime with \(q\), Chris Hall showed how to explicitly calculate \(L(T,E/K) \bmod{N}\). We present novel theorems going beyond Hall’s.