  Generalizing the work of A. Morra and the authors, we give explicit formulas
for the Dirichlet series generating function of $D_{\ell}$-extensions of odd
prime degree $\ell$ with given quadratic resolvent. Over the course of our
proof, we explain connections between our formulas and the Ankeny-Artin-Chowla
conjecture, the Ohno-Nakagawa relation for binary cubic forms, and other
topics. We also obtain improved upper bounds for the number of such extensions
(over Q) of bounded discriminant.
