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We continue the study of the spectral theory associated to integrablemetrics, started in our previous paper arXiv:1301.1793 [math.SP]. We introducethe notion of 1-integrable metric on line-bundles on a compact Riemann surface.We extend the spectral theory of generalized Laplacians to line-bundlesequipped with 1-integrable metrics. As an application, we recover the followingidentity:[\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 inarXiv:1301.1792 [math.NT].

Sur la th\'eorie spectrale des m\'etriques int\'egrables sur une surface de Riemann compacte