An explicit upper bound for $$L(1,\chi )$$ when $$\chi $$ is quadratic

Abstract

AbstractWe consider Dirichlet L-functions $$L(s, \chi )$$ L ( s , χ ) where $$\chi $$ χ is a non-principal quadratic character to the modulus q. We make explicit a result due to Pintz and Stephens by showing that $$|L(1, \chi )|\leqslant \frac{1}{2}\log q$$ | L ( 1 , χ ) | ⩽ 1 2 log q for all $$q\geqslant 2\cdot 10^{23}$$ q ⩾ 2 · 10 23 and $$|L(1, \chi )|\leqslant \frac{9}{20}\log q$$ | L ( 1 , χ ) | ⩽ 9 20 log q for all $$q\geqslant 5\cdot 10^{50}$$ q ⩾ 5 · 10 50 .