We report systematic measurements of dielectron ($e^{pm}e^{pm}$) invariant-mass $M_{ee}$ spectra at mid-rapidity in Au+Au collisions at $sqrt{s_{NN}}$ = 27, 39, and 62.4 GeV taken with the STAR detector at the Relativistic Heavy Ion Collider. For all energies studied, a significant excess yield of dielectrons is observed in the low-mass region (0.40$ < M_{ee} < 0.75$ MeV/$c^2$) compared to hadronic cocktail simulations at freeze-out. Models that include an in-medium broadening of the $rho$-meson spectral function consistently describe the observed excess. In addition, we report acceptance-corrected dielectron-excess spectra for Au+Au collisions at mid-rapidity ($left|y_{ee}right|$ $<$ 1) in the 0$-$80% centrality bin for each collision energy. The integrated excess yields for $0.4 < M_{ee} < 0.75 textrm{GeV}/c^{2}$, normalized by the charged particle multiplicity at mid-rapidity, are compared with previously published measurements for Au+Au at $sqrt{s_{NN}}$ = 19.6 and 200 GeV. The normalized excess yields in the low-mass region show no significant collision energy dependence. The data, however, are consistent with model calculations that demonstrate a modest energy dependence.
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