  Experimental data on $ R(D^{(*)}) $, $ R(K^{(*)}) $ and $ R(J/\psi) $,
provided by different collaborations, show sizable deviations from the SM
predictions. To describe these anomalies many new physics scenarios have been
proposed. One of them is leptoquark model with introducing the vector and
scalar leptoquarks coupling simultaneously to the quarks and leptons. To look
for similar possible anomalies in baryonic sector, we investigate the effects
of a vector leptoquark $U_3 (3,3, \frac{2}{3})$ on various physical quantities
related to the tree-level $ \Lambda_b \rightarrow \Lambda_c \ell ~
\overline{\nu}_\ell$ decays ($ \ell=\mu, ~\tau $), which proceed via $ b
\rightarrow c~\ell ~ \overline{\nu}_\ell$ transitions at quark level. We
calculate the differential branching ratio, forward-backward asymmetry and
longitudinal polarizations of lepton and $\Lambda_{c}$ baryon at $ \mu $ and $
\tau $ lepton channels in leptoquark model and compare their behavior with
respect to $ q^2 $ with the predictions of the standard model (SM). In the
calculations we use the form factors calculated in full QCD as the main inputs
and take into account all the errors coming from the form factors and model
parameters. It is observed that ........
