  In this work, we calculate the transition form factors of $\Lambda_b$
decaying into $\Lambda_c$ within the framework of light-cone sum rules with the
distribution amplitudes (DAs) of $\Lambda_b$-baryon. In the hadronic
representation of the correlation function, we have isolated both the
$\Lambda_c$ and the $\Lambda_c^*$ states so that the $\Lambda_b \rightarrow
\Lambda_c$ form factors can be obtained without ambiguity. We investigate the
P-type and A-type current to interpolate the light baryons for a comparison
since the interpolation current for the baryon state is not unique. We also
employ three parametrization models for DAs of $\Lambda_b $ in the numerical
calculation. We present the numerical predictions on the $\Lambda_b \rightarrow
\Lambda_c$ form factors and the branching fractions, the averaged
forward-backward asymmetry , the averaged final hadron polarization and the
averaged lepton polarization of the $\Lambda_b \to \Lambda_c \ell\mu$ decays,
as well as the ratio of branching ratios $R_{\Lambda_c}$, and the predicted
$R_{\Lambda_c}$ can be consistent with the LHCb data.
