Is there a way to show that if the Riemann hypothesis holds for Dirichlet L-function associated to primitive Dirichlet character (excluding trivial character $\chi(1)$ which could be qualified of primitive) then Riemann hypothesis holds for the Zeta function?
This is more a suggestion how to interpret the question, then a real answer. But I think it's hard to definitely answer your question, because a "good"proof of RH for any one $L$-function will most likely suggest a proof for all automorphic $L$-functions.
So here is a suggestion: Can we express zeta in terms of other primitive $L$-functions.
I don't think so, because he joint universality theorem for Dirichlet $L$-functions implies that there is not functional relation between the Riemann zeta function and any finite set of primitive Dirichlet $L$-functions, e.g,, take any finite number of primite Dirichlet character $\chi_1, \dots, \chi_n$.