The non trivial zeros of lie inside $0 \leq Re s \leq 1$ follows from the absolute convergence of the Euler product and does not need any entire function theory, not even holomorphicity. That the Riemann Zeta function is holomorphic, actually follows from the absoulute locally uniform convergence of the product.
One proof using only the languages of the adeles, harmonic analysis and no entire function theory explicitely at all was given by Ralf Meyer: http://arxiv.org/pdf/math/0311468v3
However the Fourier transform of compact supported function is always entire, so you basically just hide the complex analysis arguments.