  Using optical theorem within sophisticated fits for exclusive hadronic
productions in $e^+e^-$ collisions the interference between leptonic and
hadronic vacuum polarization functions is considered and applied for
calculation of $\mu-$pair productions below 3 GeV. Particularly, the dispersion
relation for the pion production form factor was obtained, confirming a
possible evidence for the light $\rho(1250)$ resonance. The inflated error was
introduced and used to discriminate the quality of various phenomenological
fits. Resulting hadronic vacuum polarization was compared with other
conventional sources and compared with KLOE experiment for $\mu^{-}\mu^{+}$
production at $\phi $ meson energy. Furthermore, the obtained running fine
structure coupling is compared with KLOE2 experiment for radiative return
$\mu^{-}\mu^{+}$ production at $ \omega/\rho $ meson energy and careful
analysis of the error stemming form various fits of combined data was made and
discussed.
