Non-vanishing of Dirichlet $L$-function $L(s,\chi)$ for $\Re(s)=1$

Question

I know that if $\chi$ is a non-principal Dirichlet character then the $L$-function $L(s,\chi)$ doesn't vanish for $s=1$. But, how about $s=1+it$ with $t\neq 0$? I found in this post: Zeros of Dirichlet L-functions on the line R(s)=1 in proof of Dirichlet's theorem that indeed, $L$-functions never vanish. Where can I find a proof of it? I've looked in some books and googled it, but couldn't find anything. Can sombody provide a reference? Thank you very much!