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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*}