[en] A numerical value for the running electromagnetic coupling constant in the M-bar S-bar scheme is calculated at a low-energy normalization scale equal to the τ-lepton mass M_τ. This low-energy boundary value is used for running the electromagnetic coupling constant to larger scales, where high-precision experimental measurements can be performed. Particular scales of interest are the b-quark mass for studying Υ-resonance physics and the Z-boson mass M_Z for high-precision tests of the standard model and for the determination of the Higgs-boson mass from radiative corrections. A numerical value for the running electromagnetic coupling constant at M_Z in the on-shell renormalization scheme is also given