Let $p>2$ be a prime. For $n \in \mathbb{Z}^+$ we can define \begin{equation} F(n) = (-1)^n \prod_{1<i<n, p\nmid i} i. \end{equation} Since $\mathbb{Z}$ is dense in $\mathbb{Z}_p$, we can extend $F$ uniquely to a continuous function on $\mathbb{Z}_p$, which is the $p$-adic gamma function $\Gamma_p$.

Is there any good way to explicitly compute $\Gamma_p(a/b)$ where $a,b \in \mathbb{Z}$ such that $a/b \in \mathbb{Z}_p^{\times}$? Specifically, I'm looking at $\Gamma_5(4/11)$.

A bad idea is to evaluate $F(a_n)$, where $a_n$ are the partial sums of the $5$-adic expansion of $4/11$. I don't think this would work since $F(a_n)$ becomes too difficult to compute the further you go in the expansion.

I'm aware that some values can be computed via the Gross-Koblitz formula, but I don't know how or if I can apply it to my situation.

Any help is appreciated!

**Edit:** As Henri Cohen points out, you can get Sage to do this:

```
[Input] Sage: R=Zp(5)
Sage: x = R(4/11)
Sage: x.gamma('pari')
[Output] 1 + 3*5 + 4*5^3 + 4*5^4 + 4*5^6 + 3*5^7 + 2*5^8 + 4*5^9
+ 4*5^10 + 5^11 + 4*5^13 + 5^15 + 5^16 + 3*5^17 + 2*5^18
+ 5^19 + O(5^20)
```