The exclusive process $e^+e^-rightarrowLambdabar{Lambda}$, with $Lambda to ppi^-$ and $bar{Lambda} to bar{p}pi^+$, has been studied at $sqrt{s} =$ 2.396 GeV for measurement of the $Lambda$ electric and magnetic form factors, $G_E$ and $G_M$. A data sample, corresponding to an integrated luminosity of 66.9 pb$^{-1}$, was collected with the BESIII detector for this purpose. A multi-dimensional analysis with a complete decomposition of the spin structure of the reaction enables a determination of the modulus of the ratio $R=|G_E/G_M|$ and, for the first time for any baryon, the relative phase $DeltaPhi=Phi_E-Phi_M$. The resulting values are obtained using the recent and most precise measured value of the asymmetry parameter $alpha_{Lambda}$ = $0.750~pm~0.010$ to be $R~=~0.96pm0.14~(rm stat.)pm~0.02~(rm sys.)$ and $DeltaPhi=37^{mathrm{o}}pm~12^{mathrm{o}}~(rm stat.)pm~6^{mathrm{o}}~(rm sys.)$, respectively. In addition, the cross section is measured with unprecedented precision to be $sigma = 118.7~pm~5.3~(rm stat.)pm~5.1~(rm sys.)$ pb, which corresponds to an effective form factor of $|G|=0.123~pm~0.003~(rm stat.)pm~0.003~(rm sys.)$. The contribution from two-photon exchange is found to be negligible. Our result enables the first complete determination of baryon time-like electromagnetic form factors.
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