Extremal spectral properties of the Lawson tori are studied. A Lawson torus
carries an extremal metric for some eigenvalue $\lambda_j$ of the
Laplace-Beltrami operator. We prove that the metric on a Lawson torus
$\tau_{m,k}$ is extremal for $j=2([\sqrt{m^2+k^2}]+m+k)-1.$