  In this paper, we have developed a systematic technique to regularize double
summations of Landau levels and analytically evaluated the photon vacuum
polarization at an external magnetic field. The final results are described by
Lerch transcendent $\Phi(z,s,v)$ or its $z$-derivation. We have found that the
tensor of vacuum polarization is split into not only longitudinal and
transverse parts but also another mixture component. We have obtained a
complete expression of the magnetized photon vacuum polarization at any
kinematic regime and any strength of magnetic field for the first time. In the
weak $B$-fields, after canceling out a logarithmic counter term, all three
scalar functions are limited to the usual photon polarization tensor without
turning on magnetic field. In the strong $B$-fields, the calculations under
Lowest Landau Level approximation are only valid at the region $M^2\gg
q_{\shortparallel}^2$, but not correct while $q_{\shortparallel}^2\gg M^2$,
where, an imaginary part has been missed. It reminds us, a recalculation of the
gap equation under a full consideration of all Landau Levels is necessary in
the next future.
