  We continue the study of the spectral theory associated to integrable
metrics, started in our previous paper arXiv:1301.1793 [math.SP]. We introduce
the notion of 1-integrable metric on line-bundles on a compact Riemann surface.
We extend the spectral theory of generalized Laplacians to line-bundles
equipped with 1-integrable metrics. As an application, we recover the following
identity:
[\zeta'_{\Delta_{\bar{\mathcal{O}(m)}_\infty}}(0)=T_g\bigl((\p^1,\omega_\infty);
\bar{\mathcal{O}(m)}_\infty \bigr),] obtained using direct computations in
arXiv:1301.1792 [math.NT].
