Is there a way to show that if the Riemann hypothesis holds for Dirichlet Lfunction 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 $\chi_1, \dots, \chi_n$.
Ref: http://books.google.de/books/about/The_Riemann_Zeta_Function.html?id=fNontpCu9kQC&redir_esc=y 

