  The product of the $\Lambda_b^0$ ($\overline{B}^0$) differential production
cross-section and the branching fraction of the decay $\Lambda_b^0\rightarrow
J/\psi pK^-$ ($\overline{B}^0\rightarrow J/\psi\overline{K}^*(892)^0$) is
measured as a function of the beauty hadron transverse momentum, $p_{\rm T}$,
and rapidity, $y$. The kinematic region of the measurements is $p_{\rm
T}<20~{\rm GeV}/c$ and $2.0<y<4.5$. The measurements use a data sample
corresponding to an integrated luminosity of $3~{\rm fb}^{-1}$ collected by the
LHCb detector in $pp$ collisions at centre-of-mass energies $\sqrt{s}=7~{\rm
TeV}$ in 2011 and $\sqrt{s}=8~{\rm TeV}$ in 2012. Based on previous LHCb
results of the fragmentation fraction ratio, $f_{\Lambda_B^0}/f_d$, the
branching fraction of the decay $\Lambda_b^0\rightarrow J/\psi pK^-$ is
measured to be \begin{equation*} \mathcal{B}(\Lambda_b^0\rightarrow J/\psi
pK^-)= (3.17\pm0.04\pm0.07\pm0.34^{+0.45}_{-0.28})\times10^{-4},
\end{equation*} where the first uncertainty is statistical, the second is
systematic, the third is due to the uncertainty on the branching fraction of
the decay $\overline{B}^0\rightarrow J/\psi\overline{K}^*(892)^0$, and the
fourth is due to the knowledge of $f_{\Lambda_b^0}/f_d$. The sum of the
asymmetries in the production and decay between $\Lambda_b^0$ and
$\overline{\Lambda}_b^0$ is also measured as a function of $p_{\rm T}$ and $y$.
The previously published branching fraction of $\Lambda_b^0\rightarrow J/\psi
p\pi^-$, relative to that of $\Lambda_b^0\rightarrow J/\psi pK^-$, is updated.
The branching fractions of $\Lambda_b^0\rightarrow P_c^+(\rightarrow J/\psi
p)K^-$ are determined.
