  An analysis of the astrophysical $S$ factor of the proton-proton weak capture
($\mathrm{p}+\mathrm{p}\rightarrow
{}^2\mathrm{H}+\mathrm{e}^++\nu_{\mathrm{e}}$) 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 $\Lambda$. Surprisingly, $\Lambda$ reaches a
maximum near $0.13~\mathrm{MeV}$ that has been unexplained until now. A
model-independent parametrization of $\Lambda$ valid up to about
$5~\mathrm{MeV}$ is established on the basis of recent effective-range
functions. It provides an insight into the relationship between the maximum of
$\Lambda$ and the proton-proton resonance pole at
$(-140-467\,\mathrm{i})~\mathrm{keV}$ from analytic continuation. In addition,
this parametrization leads to an accurate evaluation of the derivatives of
$\Lambda$, and hence of $S$, in the limit of zero energy.
