Axiom 3: Functional Equation

There exist numbers $Q>0$, $\alpha_j>0$, and $\Re(r_j)\ge 0$, such that

$$\Phi(s):= Q^s \prod_{j=1}^d \Gamma(\alpha_j+r_j) F(s)$$

satisfies

$$\Phi(s)=\varepsilon {\overline{\Phi}}(1-s).$$

Here $\vert\varepsilon\vert=1$ and ${\overline{\Phi}}(z)={\overline{\Phi({\overline{z}})}}$.

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