The instability of a weakly sheared density-stratified two-dimensional wavy flow to longitudinal vortices is considered. The instability mechanism is Craik–Leibovich type 2, or CL2, and the problem is posited in the context of Langmuir circulations beneath irrotational wind-driven surface waves. Of interest is the influence to the instability of Prandtl Pr and Richardson Ri numbers according to linear theory. The basis for the study is an initial value problem posed by Leibovich & Paolucci (1981) in which the liquid substrate is of semi-infinite extent and the wind-driven current is permitted to grow in the presence of neutral waves. In the present work Pr is varied from zero to infinity, and both stabilizing and destabilizing Ri are considered; so too are monochromatic and measured wave fields, and laminar and turbulent velocity profiles. Only the Ri = 0 results recover those of Leibovich & Paolucci. For stabilizing Ri, it is found in general that diminishing Pr are destabilizing to Langmuir circulations (LCs), and thus that LCs can be present or absent at the same Langmuir number La provided Ri ≠ 0. It is further found that two branches of neutral curves occur for some combinations of Pr and Ri, and that minor changes in either parameter permit the preferred spacing to switch from one branch to the other. In consequence the preferred spanwise spacing may change from smaller than the wavelength of the dominant waves to larger than it. Furthermore, although LCs will not form at inverse La below a global lower bound given by an energy stability analysis, the actual value of La at onset is found to depend greatly upon local details of the wave and shear fields. Interestingly although this global lower bound is independent of Ri and Pr for Ri [ges ] 0, that is not the case for Ri < 0, where it approaches zero as Ri → −∞, indicating that the CL2 instability is viable even at low Reynolds numbers.