Numerical analytical theory – Non-trivial real zeros of Dirichlet L functions

When dealing with the theorem of prime numbers in arithmetic progressions, the possible presence of a real zero close to $ 1 $ at most one real character mod $ q $. On the other hand, it is also known that the Riemann $ zeta $ The function does not disappear (0, 1).

Is there any result that shows that some Dirichlet L functions (associated with a non-main character) do not disappear? far since $ 1 $ ? I think it does not fade in $ 1/2 $ It is still open in general, but it may be known in some cases.