Recent analytical and numerical results for the three-loop
polarization function allow to present a phenomenological analysis
of the cross section for massive quark production in electron positron
annihilation to order $\alpha_s^2$. Numerical predictions based on fixed
order perturbation theory are presented for charm and bottom
production above 5 and 11.5 GeV, respectively. The contribution
from these energy regions to $\alpha(M_Z^2)$, the running QED coupling
constant at scale M_Z, are given. The dominant terms close to threshold,
i.e. in an expansion for small quark velocity $\beta$, are presented.
