We define a generalized Golomb-Dickman constant $λ_θ$ as the limiting expected proportion of the longest cycle in random permutations under the Ewens measure with parameter $θ> 0$. Exploiting the independence properties of Kingman's Poisson process construction of the Poisson-Dirichlet distribution, we obtain an explicit integral representation for $λ_θ$ in terms of the exponential integral. The dependence of $λ_θ$ on $θ$ reflects the transition between regimes dominated by long cycles (small $θ$) and those with many small cycles (large $θ$). We also derive the asymptotic behavior of $λ_θ$ for small and large $θ$, and illustrate our results with numerical computations and Monte Carlo simulations of the Hoppe urn. Our results can be viewed as an extension of the classical calculations of Shepp and Lloyd to the Ewens setting by relatively elementary means.