  We explicitly compute the special values of the standard $L$-function $L(s,
F_{12}, \mathrm{St})$ at the critical points $s\in\{-8, -6, -4, -2, 0, 1, 3, 5,
7, 9\}$, where $F_{12}$ is the unique (up to a scalar) Siegel cusp form of
degree $3$ and weight $12$, which was constructed by Miyawaki. These values are
proportional to the product of the Petersson norms of symmetric square of
Ramanujan's $\Delta$ and the cusp form of weight $20$ for ${\rm
SL}_2(\mathbb{Z})$ by a rational number and some power of $\pi$. We use the
Rankin-Selberg method and apply the Holomorphic projection to compute these
values. To our knowledge this is the first example of a standard $L$-function
of Siegel cusp form of degree $3$, when the special values can be computed
explicitly.
