  We continue the study of the one-loop partition function of AdS$_3$ gravity
with focus on the square-integrability condition on the fluctuating fields. In
a previous work we found that the Brown-Henneaux boundary conditions follow
directly from the $L^2$ condition. Here we rederive the partition function as a
ratio of Laplacian determinants by performing a suitable decomposition of the
metric fluctuations. We pay special attention to the asymptotics of the fields
appearing in the partition function. We also show that in the usual computation
using ghost fields for the de Donder gauge, such gauge condition is accessible
precisely for square-integrable ghost fields. Finally, we compute the spectrum
of the relevant Laplacians in thermal AdS$_3$, in particular noticing that
there are no isolated eigenvalues, only essential spectrum. This last result
supports the analytic continuation approach of David, Gaberdiel and Gopakumar.
The purely essential spectra found are consistent with the independent results
of Lee and Delay of the essential spectrum of the TT rank-2 tensor
Lichnerowickz Laplacian on asymptotically hyperbolic spaces.
