Riemann's functional equation allows $\Gamma (s)$ to be expressed as a finite composition of elememtary functions and zeta functions: $$\Gamma (1-s)=\frac{\pi ^{1-s}\zeta (s)}{2^s \sin (\pi s/2)\zeta (1-s)},\quad 1-s\in\mathbb{C}.$$
Conversely, is there an equation which allows $\zeta (s)$ to be expressed as a finite composition of elementary functions and gamma functions?