This arises from the principle of Analytic Continuation. The Gamma function exists independently of the integral representation you alluded to. It's just that integral representation is equal to the Gamma function when . That may sound confussing. Consider the canonial example: The series representation for . This series ``represents'' only when but the function exists for all values of . It's a similar situation for the Gamma function: The Gamma function is a complex-analytic function defined throughout the complex plane except for poles (singularities) at zero and the negative integers with the common integral representation of it valid only when :
where is a reverse-Hankel contour in the complex plane (there are other ways to define the Gamma function).
Even though the expression can be derived from the common integral representation of Gamma which is only valid when , because of the Principle of Analytic Continuation, the expression is still valid for all and not just s>0.