We comparatively analyze the rare $Lambda _brightarrow Lambda ell^+ ell^-$ channel in standard model, supersymmetry and Randall-Sundrum model with custodial protection (RS$_c$). Using the parametrization of the matrix elements entering the low energy effective Hamiltonian in terms of form factors, we calculate the corresponding differential decay width and lepton forward-backward asymmetry in these models. We compare the results obtained with the most recent data from LHCb as well as lattice QCD results on the considered quantities. It is obtained that the standard model, with the form factors calculated in light-cone QCD sum rules, can not reproduce some experimental data on the physical quantities under consideration but the supersymmetry can do it. The RS$_c$ model predictions are roughly the same as the standard model and there are no considerable differences between the predictions of these two models. In the case of differential decay rate, the data in the range $4 $ GeV$^2/$c$^4leq$ $q^2 leq 6$ GeV$^2/$c$^4$ can not be described by any of the considered models.
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