Let G = SL2(K) with K a local function field of characteristic 2. We review the Artin-Schreier symbol for the field K, and show that this leads to a parametrization of certain L-packets in the smooth dual of G. The L-packets in the principal series are parametrized by quadratic extensions, and the supercuspidal L-packets of cardinality 4 are parametrized by biquadratic extensions. Each supercuspidal packet of cardinality 4 is accompanied by a singleton packet for SL1(D). We provide lower bounds for the depths of the irreducible constituents of all these L-packets for SL2(K) and its inner form SL1(D).