Numerical calculations of the Landau constant are presented for the case of a shear layer of finite Reynolds number, having the velocity profile $\overline{u} = \tan h\,y$. It is found that this parameter has a strong dependence on the Reynolds number for Re [les ] 100. In particular, the Landau constant is reduced by 43% from its inviscid value when Re = 40, the latter value being typical of many experiments. This percentage, however, is based upon a calculation in which the mean-flow distortion has been neglected. A rough estimate of the latter effect indicates that it could possibly increase the value of the Landau constant sufficiently that the net influence of a finite Reynolds number would be of a smaller magnitude.