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.