These last few days, I've been wondering whether one could consider the parameters/variables $\chi$ and $s$ a Dirichlet L-function depends on as coordinates such that the pair of transformations $(\chi\mapsto\bar{\chi},s\mapsto 1-s)$ defines an involutive isometry of the considered L-function. Indeed the functional equation thereof is: $$\Lambda(\chi,s)=\omega\Lambda(\bar{\chi},1-s)$$
with $\Lambda$ the complete Dirichlet L-function and $\vert\omega\vert=1$, so that $\vert\Lambda(\chi,s)\vert=\vert\Lambda(\bar{\chi},1-s)\vert$. Hence the "covariance group" of the functional equation should be isomorphic to $Z/2Z$.
Can this be used to get information about the location of the zeroes of the complete L-function, and thus, nontrivial zeroes of the (incomplete) L-function? Has such an approach of GRH related topics been considered so far?
Many thanks in advance.