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The differential branching fraction of the rare decay $\Lambda^{0}_{b}\rightarrow \Lambda \mu^+\mu^-$ is measured as a function of $q^{2}$, thesquare of the dimuon invariant mass. The analysis is performed usingproton-proton collision data, corresponding to an integrated luminosity of $3.0\mbox{ fb}^{-1}$, collected by the LHCb experiment. Evidence of signal isobserved in the $q^2$ region below the square of the $J/\psi$ mass. Integratingover $15 < q^{2} < 20 \mbox{ GeV}^2/c^4$ the branching fraction is measured as$d\mathcal{B}(\Lambda^{0}_{b} \rightarrow \Lambda \mu^+\mu^-)/dq^2 = (1.18 ^{+0.09} _{-0.08} \pm 0.03 \pm 0.27) \times 10^{-7} ( \mbox{GeV}^{2}/c^{4})^{-1}$,where the uncertainties are statistical, systematic and due to thenormalisation mode, $\Lambda^{0}_{b} \rightarrow J/\psi \Lambda$, respectively.In the $q^2$ intervals where the signal is observed, angular distributions arestudied and the forward-backward asymmetries in the dimuon ($A^{l}_{\rm FB}$)and hadron ($A^{h}_{\rm FB}$) systems are measured for the first time. In therange $15 < q^2 < 20 \mbox{ GeV}^2/c^4$ they are found to be $A^{l}_{\rm FB} =-0.05 \pm 0.09 \mbox{ (stat)} \pm 0.03 \mbox{ (syst)}$ and $A^{h}_{\rm FB} =-0.29 \pm 0.07 \mbox{ (stat)} \pm 0.03 \mbox{ (syst)}$.

Differential branching fraction and angular analysis of $\Lambda^{0}_{b} \rightarrow \Lambda \mu^+\mu^-$ decays