A search for lepton-flavor-violating Z -> e tau and Z -> mu tau decays with pp collision data recorded by the ATLAS detector at the LHC is presented. This analysis uses 139 fb(-1) of Run 2 pp collisions at root s = 13 TeV and is combined with the results of a similar ATLAS search in the final state in which the tau lepton decays hadronically, using the same data set as well as Run 1 data. The addition of leptonically decaying tau leptons significantly improves the sensitivity reach for Z -> l tau decays. The Z -> l tau branching fractions are constrained in this analysis to B(Z -> e tau) < 7.0 x 10(-6) and B (Z -> mu tau) < 7.2 x 10(-6) at 95% confidence level. The combination with the previously published analyses sets the strongest constraints to date: B(Z -> e tau) < 5.0 x 10(-6) and B(Z -> mu tau) < 6.5 x 10(-6) at 95% confidence level.