  Let $E$ be a quadratic semisimple extension of a local field $F$ of
characteristic zero. We determine explicit relation between gamma factors for
Asai representations of $R_{E/F}{\rm GL}_{2/E}$ defined by the Weil-Deligne
representations and local zeta integrals. When $E = F\times F$, the results
were due to Henniart and Jacquet. We completed the theory in this article based
on explicit calculation.
