A note on Li's coefficients for Hecke L-functions

Recently, the Li criterion for the Riemann hypothesis has been extended for a general class of L-functions, so-called the Selberg class [S. Omar and K. Mazhouda, Le critere de Li et l'hypothese de Riemann pour la classe de Selberg, J. Number Theory 125(1) (2007) 50-58; Corrigendum et addendum a "Le critere de Li et l'hypothese de Riemann pour la classe de Selberg" [J. Number Theory 125(1) (2007) 50-58], J. Number Theory 130(4) (2010) 1109-1114]. Further numerical computations have been done to verify the positivity of some Li coefficients for the Dirichlet L-functions and the Hecke L-functions [S. Omar, R. Ouni and K. Mazhouda, On the zeros of Dirichlet L-functions, LMS J. Comput. Math. 14 (2011) 140-154; On the Li coefficients for the Hecke L-functions, Math. Phys. Anal. Geom. 17(1-2) (2014) 67-81]. Based on the latter numerical experiments, it was conjectured that those coefficients are increasing in n. In this note, we show actually that the Riemann hypothesis holds if and only if the Li coefficients for the Hecke L-functions are increasing in n.