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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  q{\mathbb{F}_{q}}   , and assume for simplicity that  q  is a prime withq  ≡    1   ⁢   (   mod  ⁢   4   ){q\equiv 1~{}(\mathrm{mod}\,4)}   . We compute the second and third moment ofL  ⁢   (   1  /  2   ,   χ  D   ){L(1/2,\chi_{D})}   , when  D  is a monic square-free polynomial of degree2  ⁢  g   +  1{2g+1}   , asg  →  ∞{g\to\infty}   . The answer we get for the second moment agrees with Andrade and Keating’s conjectured formula in [ 4 ]. For the third moment, we check that the leading term agrees with the conjecture.

The second and third moment of L (1/2,χ) in the hyperelliptic ensemble