Lower Bound Of Dirichlet L-function For Real Primitive Characters At S=1

In this paper we improve the results of Siegel-Tatuzawa theorem based on others' works. The main way is Hoffstein's which was progressed by professor HongwenLu and professor Chungang-Ji in their papers. We also use arithmetic property of quadratic field, double quadratic field and others' new results. It also improves the upper bound of real zeros which are possible in L-function L(s,χ) at closeness of s=1 for real primitive characters.It has two main results in this thesis. One is that let positive constantε＜1/(6 log 10),χbe a real primitive character modulo k such thatκ＞e1╱ε. Then with at most one exception the following holds.If we suppose the Riemann hypothesis holds and the above conditions doesn'tchange, then the second result can hold. It's as follows.