Derivation of Freund–Witten adelic formula for four-point Veneziano amplitudes

On the base of analysis on the adelic group (Teyte Tate's formula) a regularization is proposed for the divergent infiniteproduct of $p$-adic $\Gamma$-functions
$$\Gamma _p(\alpha )=\frac {1-p^{\alpha -1}}{1-p^{-\alpha }}\,, \quad p=2,3,5,\dots \,.$$
Adelic formula
$$\,{\operatorname {reg}}\,\prod _{p=2}^\infty \Gamma _p(\alpha )=\frac {\zeta (\alpha )}{\zeta (1-\alpha )},$$
($\zeta (\alpha )$ is Riemann $\zeta$-function) is proved.