We investigate the influence of the strong Coulomb field of a heavy nucleus on massive neutrinos, produced in the K–shell electron capture (EC) decays of the H–like 140 Pr 58+ and 142 Pm 60+ ions. The corrections to the neutrino masses due to virtually produced charged lepton W–boson pairs in the strong Coulomb field of a nucleus with charge Ze are calculated and discussed with respect to their influence on the period of the time–modulation of the number of daughter ions, observed recently in the EC–decays of the H–like 140 Pr 58+ and 142 Pm 60+ ions at GSI in Darmstadt. These corrections explain the 2.75 times higher difference of the squared neutrino masses obtained from the time–modulation of the EC–decays with respect to the value deduced from the neutrino–oscillation experiments of KamLAND. The values of neutrino masses are calculated. PACS: 12.15.Ff, 13.15.+g, 23.40.Bw, 26.65.+t The experimental investigation of the EC
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