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This work deals with fractional calculus of complex functions. More precisely, we study the case of zeta functions. In fact, this family of functions seems to be of independent interest with respect to fractional calculus. Indeed, the fractional derivative of zeta functions is given by its integer derivative by simply interchanging integer and fractional derivative orders. The same holds for many other properties of this function family (e.g., functional equation, zero-free regions, etc.). Moreover, fractional calculus of zeta functions can be extended to Banach algebras and function spaces, showing its relevant role in pure and applied mathematics.