L-VALUES OF HARMONIC MAASS FORMS

Bruinier, Funke, and Imamoglu have proved a formula for whatcan philosophically be called the "centralL-value" of the modularj-invariant.Previously, this had been heuristically suggested by Zagier. Here, we interpretthis "L-value" as the value of an actualL-series, and extend it to all integralarguments and to a large class of harmonic Maass forms which includes allweakly holomorphic cusp forms. The context and relation to previously de-finedL-series for weakly holomorphic and harmonic Maass forms are discussed.These formulas suggest possible arithmetic or geometric meaning ofL-valuesin these situations. The key ingredient of the proof is to apply a recent theoryof Lee, Raji, and the authors to describe harmonic MaassL-functions usingtest functions