  We determine the strong coupling constant $\alpha_s$ from a lattice
calculation of vacuum polarization functions (VPF) in three-flavor QCD with
dynamical overlap fermions. Fitting lattice data of VPF to the continuum
perturbative formula including the operator product expansion, we extract the
QCD scale parameter $\Lambda_{\overline{MS}}^{(3)}$. At the $Z$ boson mass
scale, we obtain $\alpha_s^{(5)}(M_Z)=0.1181(3)(^{+14}_{-12})$, where the first
error is statistical and the second is our estimate of various systematic
uncertainties.
