Are there any set of axioms that completely characterize the Riemann zeta function?
 i.e. like Ressayre axioms for the exponential function in an exponential field or functional equations.



Axioms for Riemann $\zeta$ function
Perhaps you are looking for something like *Hamburger's Theorem*? 


It states, essentially, that the only Dirichlet series with a finite number of singularities satisfying the same functional equation as the zeta-function is the zeta-function. You can find the details in Titchmarsh's book. 


Googling I found the following link: <http://www.mat.univie.ac.at/~esiprpr/Zetaproc/patterson.pdf>


