The second and third moment of $L(\frac{1}{2},\chi )$ in the hyperelliptic ensemble

We obtain asymptotic formulas for the second and third moment of quadratic Dirichlet $L$–functions at the critical point, in the function field setting. We fix the ground field ${\mathbb{F}}_q$, and assume for simplicity that $q$ is a prime with $q\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}4)$. We compute the second and third moment of $L(1\mathrm{/}2,{\chi }_D)$ when $D$ is a monic, square-free polynomial of degree $2g+1$, as $g\to \mathrm{\infty }$. The answer we get for the second moment agrees with Andrade and Keating’s conjectured formula. For the third moment, we check that the leading term agrees with the conjecture.