Let $F$ be a local field (whose residue field is $q$) and $E$ its quadratic extension. Let $\pi$ be a irreducible principal series representation $\pi(\chi_1, \chi_2)$ of $GL_F(2)$ especially where $\chi_1, \chi_2$ are unitary characters. Then I know $L_F(s,\pi)=\frac{1}{1-\chi_1(\varpi)q^s} \cdot \frac{1}{1-\chi_2(\varpi)q^s}$. Then what is $L_E(s,BC(\pi))$ in terms of $\chi_1, \chi_2$?