  We show that Lyapunov exponent for the Sinai billiard is $\lambda =
-2\log(R)+C+O(R\log^2 R)$ with $C=1-4\log 2+27/(2\pi^2)\cdot \zeta(3)$ where
$R$ is the radius of the circular scatterer. We consider the disk-to-disk-map
of the standard configuration where the disks is centered inside a unit square.
