An analysis of the astrophysical S factor of the proton-proton weak capture (p+p → 2 H+e + +νe) is performed on a large energy range covering solar-core and early Universe temperatures. The measurement of S being physically unachievable, its value relies on the theoretical calculation of the matrix element Λ. Surprisingly, Λ reaches a maximum near 0.13 MeV that has been unexplained until now. A model-independent parametrization of Λ valid up to about 5 MeV is established on the basis of recent effective-range functions. It provides an insight into the relationship between the maximum of Λ and the proton-proton resonance pole at (−140 − 467 i) keV from analytic continuation. In addition, this parametrization leads to an accurate evaluation of the derivatives of Λ, and hence of S, in the limit of zero energy. * FNRS Aspirant, dgaspard@ulb.ac.be