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This is a continuation of our previous paper published in Israel J. Math 187(2012), 317-369. The aim of the paper here is to study the Fourier coefficientsof Arakawa lifts in relation with central values of automorphic L-functions. Inthe previous paper we provide an explicit formula for the Fourier coefficientsin terms of toral integrals of automorphic forms on multiplicative groups ofquaternion algebras. In this paper, after studying explicit relations betweenthe toral integrals and the central $L$-values, we explicitly determine theconstant of proportionality relating the square norm of a Fourier coefficientof an Arakawa lift with the central L-value. We can relate the square norm withthe central value of some $L$-function of convolution type attached to the liftand a Hecke character. We also discuss the existence of strictly positivecentral values of the L-functions in our concern.

Fourier expansion of Arakawa lifting II: Relation with central L-values