In recent papers by D'Eath and Esposito two kinds of boundary conditions, local and nonlocal (spectral), were used to study the contribution of fermions to the one-loop prefactor in the Hartle-Hawking wave function of the Universe. Using the $\ensuremath{\zeta}$-function technique they found that for the case of massless Majorana fermions on a flat background bounded by a three-sphere the values $\ensuremath{\zeta}(0)$ coincide for the two kinds of boundary conditions mentioned above. Implementing our version of $\ensuremath{\zeta}$ regularization elaborated earlier, we calculate $\ensuremath{\zeta}(0)$ for both the massive and massless fermions on the background representing the part of the four-dimensional de Sitter sphere bounded by a three-sphere. For the massless fermions our results coincide with results for a flat background and, consequently, the results for both types of boundary conditions are the same. However, for massive fermions the values $\ensuremath{\zeta}(0)$ for local and spectral boundary conditions differ on the de Sitter background and on the flat one as well.