  It is known that the second Leibniz homology group $HL_2(stl_n(R))$ of the
Steinberg Leibniz algebra $stl_n(R)$ is trivial for $n\geq 5$. In this paper,
we determine $HL_2(stl_n(R))$ explicitly (which are shown to be not necessarily
trivial) for $n=3, 4$ without any assumption on the base ring.
