We consider zeta functions ζ(s; \( \mathfrak{a} \) ) given by Dirichlet series with multiplicative periodic coefficients and prove that, for some classes of functions F , the functions F(ζ(s; \( \mathfrak{a} \) )) have infinitely many zeros in the critical strip. For example, this is true for sin(ζ(s; \( \mathfrak{a} \) )).
