$$ G_{1,4}^{2,0}\left(z\left|\begin{smallmatrix}1\\ 0,0,\frac{1}{2},\frac{1}{2}\end{smallmatrix}\right.\right) $$
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This is what I got from Mathematica $$G_{1,4}^{2,0}\left(z\left| \begin{array}{c} 1 \\ 0,0,\frac{1}{2},\frac{1}{2} \\ \end{array} \right.\right)=-\frac{1}{\pi}\left[3\gamma -4 z \, _2F_4\left(1,1;\frac{3}{2},\frac{3}{2},2,2;-z\right)+\log 16z \right]$$
Hope it helps
Edit.
Mathematica code
MeijerG[{{},{1}},{{0,0},{1/2,1/2}},z]//FunctionExpand
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