  A theory of non-local linear ion-temperature-gradient (ITG) drift modes while
retaining non-adiabatic electrons is presented, extending the previous work [S.
Moradi, et al {\em Phys. Plasmas} {\bf 18}, 062106 (2011)]. A dispersion
relation is derived to quantify the effects of the fractional velocity operator
in the Fokker-Planck equation modified by temperature gradients and
non-adiabatic electrons on the real frequency and growth rate. Solving the
dispersion relation, it is shown here that as the plasma becomes more
turbulent, it deviates from a Maxwellian distribution and becomes L\'{e}vy
distributed. The resulting L\'{e}vy distribution of the plasma may thus
significantly alter the transport. The relative effect of the fractional
derivative is larger on the real frequency than on the growth rate of the ITG
mode.
