Laplacians of fullerenes (C-42-C-90), which are quite important in the characterization of structures, enumeration of spanning trees, computation of the magnetic properties of fullerenes, computation of the vibrational spectra, and so on, are obtained using high-precision computational algorithms. The spectra and the characteristic polynomials of the Laplacians of fullerenes are computed. From the related Riemann xi functions of fullerenes, the number of spanning trees of fullerenes is computed. Several interesting results pertinent to the spectra of Laplacians of fullerenes are outlined.