  Let $G$ be a connected reductive group over $\mathbb{Q}$. In this paper, we
will stabilize the local trace formula, in particular, we construct the
explicit form of the spectral side of stable local trace formula in the
Archimedean case, when one component of the test function is cuspidal. Then we
will also give the multiplicity formula for discrete series. At the same time,
we obtain the stable version of $L^{2}$-Lefschetz number formula.
