Gamma factors for the Asai representation of GL(2)

Let E be a quadratic semisimple extension of a local field F of characteristic zero. We determine explicit relation between the gamma factors for the Asai representation of R _E/F GL _2/E defined by the Weil-Deligne representations and the local zeta integrals. When E=F×F, the results were due to Henniart and Jacquet. We completed the theory in this article based on explicit calculation.