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The Hardy spaces of Dirichlet series, denoted by ${}^p$ (p ≥ 1), have been studied by Hedenmalm et al. (1997) when p = 2 and by Bayart (2002) in the general case. In this paper we study some $L^p$-generalizations of spaces of Dirichlet series, particularly two families of Bergman spaces, denoted ${}^p$ and ${ℬ}^p$. Each could appear as a “natural” way to generalize the classical case of the unit disk. We recover classical properties of spaces of analytic functions: boundedness of point evaluation, embeddings between...