Recent experimental results of R(D(*)) deviate from the standard model (SM) by 3.1σ, suggesting a new physics (NP) that affects the b→cτν¯τ transition. Motivated by this, we investigate the possible NP effects in the Λb→Λcτν¯τ decay. For this purpose, assuming the neutrinos are left-handed, we calculate in detail the helicity amplitudes of Λb→Λcℓν¯ℓ (ℓ=e−, μ−, and τ−) decays with all possible four-fermion operators. Within the latest results of Λb→Λc form factors from lattice QCD calculations, we study these decays in a model-independent manner. The differential and total branching fractions, the longitudinal polarizations of final leptons and hadrons, the forward-backward asymmetries in the lepton side, the convexity parameters, and the ratio R(Λc) are calculated. In SM, we obtain the ratio R(Λc)=0.33±0.01. Supposing that NP only affects the third generation fermions, we present the correlations among R(D), R(D*), and R(Λc), as the Λb→Λcτν¯τ and B→D(*)τν¯τ are all induced by b→cℓν¯ℓ. We perform a minimum χ2 fit of the Wilson coefficient of each operator to the latest experimental data of different observables, including the ratios R(D(*)) and R(J/ψ) and the τ polarization Pτ(D*). It is found that the left-handed scalar operator OSL affects the branching fraction remarkably, and the ratio R(Λc) can be enhanced by 30%. For other operators, the ratio amounts to 0.38±0.02, which is larger than prediction of SM by 20%. Using the fitted values of the Wilson coefficients of the single NP operators, we also give a prognosis for the physical observables of Λb→Λcτν¯τ, including the ratio R(Λc), forward-backward asymmetry, and other polarized observables as well as the differential branching fraction. Furthermore, we also study the effects of three typical NP models on the ratio and the differential branching fraction of Λb→Λcτν¯τ. We hope our results can be tested in the current LHCb experiment and the future high energy experiments.