This paper presents an analysis of the Λ0b→ J/ψΛ angular distribution and the transverse production polarisation of Λ0b baryons in proton-proton collisions at centre-of-mass energies of 7, 8 and 13 TeV. The measurements are performed using data corresponding to an integrated luminosity of 4.9 fb−1, collected with the LHCb experiment. The polarisation is determined in a fiducial region of Λ0b transverse momentum and pseudorapidity of 1 < pT< 20 GeV/c and 2 < η < 5, respectively. The data are consistent with Λ0b baryons being produced unpolarised in this region. The parity-violating asymmetry parameter of the Λ → pπ− decay is also determined from the data and its value is found to be consistent with a recent measurement by the BES III collaboration.