The product of the $Lambda_b^0$ ($overline{B}^0$) differential production cross-section and the branching fraction of the decay $Lambda_b^0rightarrow J/psi pK^-$ ($overline{B}^0rightarrow J/psioverline{K}^*(892)^0$) 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 $Lambda_b^0rightarrow J/psi pK^-$ is measured to be begin{equation*} mathcal{B}(Lambda_b^0rightarrow J/psi pK^-)= (3.17pm0.04pm0.07pm0.34^{+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 $overline{B}^0rightarrow J/psioverline{K}^*(892)^0$, 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 $Lambda_b^0rightarrow J/psi ppi^-$, relative to that of $Lambda_b^0rightarrow J/psi pK^-$, is updated. The branching fractions of $Lambda_b^0rightarrow P_c^+(rightarrow J/psi p)K^-$ are determined.
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