Recent observations constrained the tangential velocity of M31 with respect to the Milky Way to be vM31, tan < 34.4 km s−1and the radial velocity to be in the range vM31, rad = −109 ± 4.4 km s−1. In this study we use a large volume high-resolution N-body cosmological simulation (Bolshoi) together with three constrained simulations to statistically study this kinematics in the context of the Λ cold dark matter (ΛCDM). The comparison of the ensembles of simulated pairs with the observed Local Group (LG) at the 1σ level in the uncertainties has been done with respect to the radial and tangential velocities, the reduced orbital energy (etot), angular momentum (lorb), and the dimensionless spin parameter, λ. Our main results are (1) the preferred radial and tangential velocities for pairs in ΛCDM are vr = −80 ± 20 km s−1 and vt = 50 ± 10 km s−1, (2) pairs around that region are 3–13 times more common than pairs within the observational values, (3) 15%–24% of LG-like pairs in ΛCDM have energy and angular momentum consistent with observations, while (4) 9%–13% of pairs in the same sample show similar values in the inferred dimensionless spin parameter. It follows that within current observational uncertainties the quasi-conserved quantities that characterize the orbit of the LG, i.e., etot, lorb, and λ, do not challenge the standard ΛCDM model, but the model is in tension with regard to the actual values of the radial and tangential velocities. This might hint to a problem of the ΛCDM model to reproduce the observed LG.