What function satisfies the following equation? $$f(x)e^{-x}\Gamma(x/\pi)=f(\pi/2-x)e^{x-\pi/2}\Gamma(1/2-x/\pi)$$
I think it should be similar to Zeta function, but what is it exactly?

You are right about the similarity of $f(x)$ to the Riemann Zeta function. 
Using the following identity that can be found here
$$\pi^{-\frac{z}{2}}\Gamma\left(\frac{z}{2}\right)\zeta(z)=\pi^{-\frac{1-z}{2}}\Gamma\left(\frac{1-z}{2}\right)\zeta(1-z)$$
and setting $z=\frac{2x}{\pi}$ results in
$$\color{red}{\pi^{-\frac{x}{\pi}}}\Gamma\left(\frac{x}{\pi}\right)\color{red}{\zeta\left(\frac{2x}{\pi}\right)}=\color{red}{\pi^{-\frac{1}{2}-\frac{x}{\pi}}}\Gamma\left(\frac{1}{2}-\frac{x}{\pi}\right)\color{red}{\zeta\left(1-\frac{2x}{\pi}\right)}$$
Comparing this expression (paying particular attention to the parts highlighted in red) with the relationship given in the question, we can deduce that
$$f(x)=\pi^{-\frac{x}{\pi}}\zeta\left(\frac{2x}{\pi}\right)e^{x}$$
