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 $q \equiv 1 \pmod 4$. We compute the second and third moment of $L(1/2,\chi_D)$ when $D$ is a monic, square-free polynomial of degree $2g+1$, as $g \to \infty$. 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.