Calculation of $\bar{\alpha}_{\rm Q.E.D.}$ on the Z

We perform a new, detailed calculation of the hadronic contributions to the running electromagnetic coupling, $\bar{\alpha}$, defined on the Z particle (91 GeV). We find for the hadronic contribution, including radiative corrections, $$10^5\times \deltav_{\rm had.}\alpha(M_Z^2)= 2740\pm12,$$ or, excluding the top quark contribution, $$10^5\times \deltav_{\rm had.}\alpha^{(5)}(M_Z^2)= 2747\pm12.$$ Adding the pure QED corrections we get a value for the running electromagnetic coupling of $$\bar{\alpha}_{\rm Q.E.D.}(M_Z^2)= {{1}\over{128.965\pm0.017}}.$$