Special values of Meijer G-Function

We define a function $$f(k)= G_{3,3}^{3,2}\left(1\left| \begin{array}{c} -1,\frac{2}{k}-2,\frac{1}{k}-1 \\ 0,\frac{1}{k}-2,\frac{2}{k}-2 \\ \end{array} \right.\right)$$ where $$G(\cdot)$$ denote Meijer G-Function.

I noticed that for $$k=2,3,4,6$$, $$f(k)$$ has a simple form $$f(2)= \frac{1}{4} \left(4+\pi ^2\right), \quad f(3)= \frac{2 \pi \left(2 \pi \sqrt{3}+3\right)}{9 \sqrt{3}}, \quad f(4)=\frac{1}{4} \pi (2+3 \pi ), \quad f(6)=\frac{\pi \left(5 \pi \sqrt{3}+4\right)}{3 \sqrt{3}}$$ But I could not figure out what $$f(5)$$ should be.

Does $$f(5)$$ also has a simple form?

• Mathematica/Wolfram Dev Platform has that function. Have you tried the Simplify function on it? – Adrian Keister Apr 26 at 18:55
• Could you give me the Mahematica input form of $f(k)$. I am going to university today and they have the CAS. I could try. – Claude Leibovici Apr 27 at 4:50
• Thanks, but l have already tried Mathematica. Now I believe there probably isn't a simpler formula for f(5) – ablmf Apr 27 at 6:35