Analytic continuation of $\zeta_3(s,k)$ in the critical strip. Arithmetical part

In this paper we study the zeta-function
$$ \zeta_3(s,k)=\sum^\infty_{n=1}\frac{\tau_3(n)\tau_3(n+k)}{(\sqrt{n(n+k)})^s} $$
with the help of the technique of automorphic functions for $SL_3(Z)$.