Additional-term conjecture for moments of quadratic Dirichlet L-functions

Let $\EuScript F_{\EuScript D}$ be the family of real primitive quadratic Dirichlet characters $\chi_d$, with $d$ nonzero and squarefree, and let $r\geq4$. For integers $N\geq1$ and real $\Theta$ satisfying $(N+1)^{-1}<\Theta<N^{-1}$, consider the $r$th moment over $|d|<D$. Additional-term conjecture. As $D\to\infty$, there are polynomials $Q_{n,r}(x)$ such that

$$\sum_{\substack{\chi_d\in\EuScript F_{\EuScript D}\\\\ |d|<D}}L(\tfrac12,\chi_d)^r =\sum_{n=1}^ND^{1/2+1/(2n)}Q_{n,r}(\log D)+O\bigl(D^{(1+\Theta)/2}\bigr).$$

This asserts lower-order terms beyond the leading moment term, an expectation motivated by prior work and partially supported by smoothed third-moment results; the stated higher-moment formula is presented as a conjectural pattern.