The authors develop a calculational technique for zeta -functions of differential operators on manifolds with boundaries with explicitly unknown spectra and apply it for the computation of the 1-loop contributions of matter fields to the Hartle-Hawking wavefunction of the universe. zeta (0) on the background of the Euclidean de Sitter manifold bounded by a three-dimensional sphere is obtained for scalar, electromagnetic, graviton, spin-1/2 and spin-3/2 fields, thus generalizing the results previously known for the case of flat spacetime.