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?