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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.

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Axioms for Riemann $\zeta$ function

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