  In our previous work, based on the $SU(6)$ spin-flavor wave function, we
regard $\Lambda$ and $p$ as composed of different quark-diquark configurations
and established the Bethe-Salpeter (BS) equations of configurations for quark
and scalar diquark. In our present work, we apply this model to calculate the
form factors of the semileptonic transitions $\Lambda_b\rightarrow\Lambda
l^+l^-$ $(l=\mu,e,\tau)$ and $\Lambda_b\rightarrow p l\bar{\nu}$ within the
Standard Model (SM). The decay $\Lambda_b\rightarrow\Lambda \mu^+\mu^-$ is
especially interesting since it has been measured in CDF and LHCb
Collaborations and this rare decay is very sensitive to new physics effects.
The decay $\Lambda_b\rightarrow p l\bar{\nu}$ is a promising mode for the
measurement of the Cabibbo-Kobayashi-Maskawa matrix element $|V_{ub}|$ at the
Large Hadron Collider. In our calculations, depending on the ranges of the
parameters in the model including the diquark mass and the interaction strength
between the quark and the diquark in the kernel of the BS equation, we find
that the branching ratio of $\Lambda_b\rightarrow\Lambda\mu^+\mu^-$ in our
model is consistent with the experimental data and the current experimental
results from LHCb agree with the differential branching ratio of
$\Lambda_b\rightarrow\Lambda\mu^+\mu^-$ from our calculation except at the
lager momentum transfer region. This indicates that there is still room for
possible new physics effects. We also give comparisions of the total branching
ratios of $\Lambda_b\rightarrow\Lambda l^+l^-$ and $\Lambda_b\rightarrow p
l\bar{\nu}$ with those given by other phenomenological methods.
