  A method combining the Lagrange-mesh and the complex Kohn variational methods
is developed for computing the $\mathcal{S}$ matrix of a 2$+$1 elastic
scattering in the frame of three-body Coulomb systems. Resonance parameters can
be obtained from values of the $\mathcal{S}$ matrix at several scattering
energies. The method is illustrated with the $S$-wave low-energy scattering of
an electron onto hydrogen. The computed phase shifts are at least as accurate
as the literature results, and the resonance parameters are more accurate than
the best literature results by several orders of magnitude. Both the infinite
and finite proton mass cases are considered.
