Not a closed form answer, but this can be solved as a semidefinite program. In particular, we can rewrite the task as
\begin{equation*}
\min_{t\ge 0, s}\ t\quad \text{s.t.}\quad (A-sI)^*(A-sI) \preceq tI,
\end{equation*}
which results in the SDP
\begin{equation*}
\min_{s,t}\ t\quad \text{s.t.}\quad \begin{bmatrix} tI & A-sI\\ (A-sI)^* & tI\end{bmatrix} \succeq 0.
\end{equation*}