I see, I was running the same line in Mathematica and received the same output. But I still have the same issue in Matlab. Do you have a reference for the third formula? The documentation of Mathematica says $\bar{F}_1(a,b;c;x)=F_1(a,b;c;x)/\Gamma(c)$ which gives me a different but also wrong result

Thank you for adding more explainations. However I wasn't able to verify your transformation from the first to the second formula for $Q^0_\nu(z)$ by implementation, is it possible that there is a typo?

Thank you very much. Do you know where I can find more explanatory material on the implementation of Mathematicas LegendreQ. I would like to understand where the real and imaginary part is coming from.