• elementary-methods
• L-functions
• zeros

Let L(s,\chi_{D}) be a Dirichlet L-function belonging to the real primitive character \chi_{D} modulus D satisfying \chi_{D}(-1)=-1. Let h(-D) be the class number of the imaginary quadratic field Q(\sqrt{-D}).
Two conjectures involving the class number h(-D) of the imaginary quadratic field belonging to the fundamental discriminant -D<0 were raised by Gauss, who published them in 1801. The first problem was about determining all the negative fundamental discriminants with class number one. The second problem was about proving that h(-D) goes to infinity as D goes to infinity.
The aim of this work is to further investigate both the Deuring Phenomenon and the Heilbronn Phenomenon. As a result, we will find better estimates regarding the influence of zeros of \zeta(s) on the exceptional zeros, and that of the non-trivial zeros of arbitrary $L$-functions belonging to non-principal characters on the exceptional zeros, respectively.