It is predicted, as a consequence of a new correction to the Dirac equation, that the electron gyromagnetic ratio gL associated with the orbital angular momentum L, is less than one. Calculations to first-order approximations show that gL = 1 - d, where d = 0.0016. This may be contrasted with an earlier observation that to first-order approximations, the electron gyromagnetic ratio corresponding to the spin magnetic moment is gs = 2(1 + d). It is emphasized that d, the anomalous part of the gyromagnetic ratios, calculated using the corrected Dirac equation is independent of the nuclear mass, and is therefore applicable to both atomic and unbound electrons. An experimental test of this new result is urged, in view of its physical significance for the relativistic quantum theories.