Infinite Product of the Riemann auxiliary function

We obtain the product for the auxiliary function $(s)$ and study some related functions as its phase $\omega (t)$ at the critical line. The function $\omega (t)$ determines the zeros of $\zeta (s)$ on the critical line. We study the influence of the zeros of $(s)$ on $\omega (t)$. Thus, the relationship between the zeros of $(s)$ and those of $\zeta (s)$ is determined.