  For any compact riemannian surface of genus three $(\Sigma,ds^2)$ Yang and
Yau proved that the product of the first eigenvalue of the Laplacian
$\lambda_1(ds^2)$ and the area $Area(ds^2)$ is bounded above by $24\pi$. In
this paper we improve the result and we show that
$\lambda_1(ds^2)Area(ds^2)\leq16(4-\sqrt{7})\pi \approx 21.668\,\pi$. About the
sharpness of the bound, for the hyperbolic Klein quartic surface numerical
computations give the value $\approx 21.414\,\pi$.
