A shortened equation for convolutions

The zeta functions of convolutions are Dirichlet series of the general form $\sum^\infty a_n\cdot n^{-s}$ therefore, they are well convergent in the right half-plane $\operatorname{Re}s>1$. In the critical strip $\operatorname{re}s\in(0,1)$ the convolutions can be represented in terms of the Linnik–Selberg zeta functions whose coefficients are Kloosterman sums. In the present paper, these two representations are combined into a single representation in the same way as the shortened equation for the classical Riemann zeta function. Bibliography: 10 titles.