On the Li Coefficients for the Hecke L-functions

In this paper, we compute and verify the positivity of the Li coefficients for the Hecke L-functions using an arithmetic formula established in Omar and Mazhouda, J. Number Theory 125(1), 50–58 (2007) and J. Number Theory 130(4), 1098–1108 (2010) and the Serre trace formula. Additional results are presented, including new formulas for the Li coefficients and a formulation of a criterion for the partial Riemann hypothesis. Basing on the numerical computations made below, we conjecture that these coefficients are increasing in n.