A search for the Standard Model (SM) Higgs boson decaying into a pair of leptons is reported. The analysis, exploiting each of the \tlep\tlep, \tlep\thad and \thad\thad final states, is based on data samples of proton-proton collisions collected by the ATLAS experiment at the LHC and corresponding to integrated luminosities of 4.6\,\ifb and 13.0\,\ifb at centre-of-mass energies of \TeV{} and 8\,\TeV{}, respectively. The observed (expected) upper limit at 95 CL on the cross-section times the branching ratio for SM is found to be 1.9 (1.2) times the SM prediction for a Higgs boson with mass =125~GeV. For this mass, the observed (expected) deviation from the background-only hypothesis corresponds to a local significance of 1.1 (1.7) standard deviations.