  Let $F$ be a non-Archimedean local field with finite residual field of
characteristic $p$. In this article we calculate the $\epsilon$-factor of pairs
for $\GL_l(F) \times \GL_{l'}(F)$ where $l$ and $l'$ are distinct primes
including the case $l=p$. For this calculation, we use the local Langlands
correspondence and non-Galois base change lift. This method leads to the
explicit conjecture of the $\epsilon$-factor of the representations of $\GL_m
\times \GL_n$ when $n$ is relatively prime to $m$ and $p$.
