  With data samples collected with the BESIII detector at seven energy points
at $\sqrt{s}= 3.68 - 3.71$ GeV, corresponding to an integrated luminosity of
333 pb$^{-1}$, we present a study of the $\Lambda$ transverse polarization in
the $e^+e^-\to\Lambda\bar\Lambda$ reaction. The significance of polarization by
combining the seven energy points is found to be 2.6$\sigma$ including the
systematic uncertainty, which implies a non-zero phase between the transition
amplitudes of the $\Lambda\bar\Lambda$ helicity states. The modulus ratio and
the relative phase of EM-$psionic$ form factors combined with all energy points
are measured to be $R^{\Psi} =$ 0.71$^{+0.10}_{-0.10}$ $\pm$ 0.03 and
$\Delta\Phi^{\Psi}$ = (23$^{+8.8}_{-8.0}$ $\pm$ 1.6$)^\circ$, where the first
uncertainties are statistical and the second systematic.
