I was just looking at Euler's reflection formula for Gamma function which states $$\Gamma (1-z)\Gamma (z)=\frac { \pi }{ \sin { (z\pi ) } } $$ but it seems to me that one more reflection formula could exist so that $$\Gamma (c+x)=\Gamma (c-f\left( x \right) )?$$ where $c\approx 1.46163$ the local minimum for Gamma function where x>0.
Our first impressions would be that $$f\left( x \right) >f(y),\quad x>y\\ f(0)=0\\ f(2-c)=c-1\\ Im(f(x))=[0,c>$$.
Can such function exist, and still follow Euler's reflection formula?