We present results about the first moment of L-functions associated to cubic characters over \(\mathbb{F}_q(T)\) when q is congruent to 1 modulo 3. The case of number fields was considered in previous work, but never for the full family of cubic twists over a field containing the third roots of unity. We will explain how to obtain an asymptotic formula with a main term, which relies on cancellation in averages of cubic Gauss sums over functions fields. We will also discuss the case q congruent to 2 modulo 3.

This is joint work with C. David and A. Florea.