| Lazarsfeld-Mukai bundles have proved useful in Brill-Noether theory of curves on K3 surfaces. Given an ample line bundle L on a K3 surface S, we study the L-slope stability of rank 3 Lazarsfeld-Mukai bundles associated to nets of type g^2_d on curves C in the linear system defined by L. When d is large enough and C is general, we obtain a dimensional statement for the variety G^2_d(C). If the Brill-Noether number is negative, we prove that any g^2_d is contained in a linear series which is induced from a line bundle on S. Some applications towards higher rank Brill-Noether theory and transversality of Brill-Noether loci are then discussed. |