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?