The definition of the $p$-adic Gamma function $\Gamma_p(x)$ for an odd prime number $p$ can be found in the book "A Course in $p$-adic analysis" by A. M. Robert. While the construction of $\log \Gamma_p(x)$ is also included in the book. The function $\log \Gamma_p(x)$ is odd, and has a series expansion in the variable $x$. For more details, see Chapter 7 of the book. \begin{equation} \log \Gamma_p(x)=(\log \Gamma_p)^{(1)}(0)x+\frac{1}{3!}(\log \Gamma_p)^{(3)}(0)x^3+\frac{1}{5!}(\log \Gamma_p)^{(5)}(0)x^5+\cdots. \end{equation} Since $\log \Gamma_p(x)$ is an odd function, we have \begin{equation} (\log \Gamma_p)^{(2n)}(0)=0. \end{equation} I am wondering whether there is an explicit method to evaluate the coefficient $(\log \Gamma_p)^{(s)}(0)$, at least the first several terms $(\log \Gamma_p)^{(3)}(0)$, $(\log \Gamma_p)^{(5)}(0)$, and $(\log \Gamma_p)^{(7)}(0)$?

One helpful observations is that since we have \begin{equation} \Gamma_p(0)=1, \end{equation} so $(\log \Gamma_p)^{(s)}(0)$ can be computed by \begin{equation} \Gamma_p^{'}(0),\Gamma^{''}_p(0),\Gamma^{'''}_p(0), \cdots,\Gamma^{(s)}_p(0). \end{equation} So this question is equivalent to evaluate $\Gamma_p^{'}(0),\Gamma^{''}_p(0),\Gamma^{'''}_p(0),\cdots$. Does anyone know an explicit formula to compute them?