Conjectures and experiments concerning the moments of $L(1/2,\chi_d)$

We report on some extensive computations and experiments concerning the moments of quadratic Dirichlet $L$-functions at the critical point. We computed the values of $L(1/2,\chi_d)$ for $- 5\times 10^{10} < d < 1.3 \times 10^{10}$ in order to numerically test conjectures concerning the moments $\sum_{|d|<X} L(1/2,\chi_d)^k$. Specifically, we tested the full asymptotics for the moments conjectured by Conrey, Farmer, Keating, Rubinstein, and Snaith, as well as the conjectures of Diaconu, Goldfeld, Hoffstein, and Zhang concerning additional lower terms in the moments. We also describe the algorithms used for this large scale computation.