  Let $F$ be a $p$-adic field. If $\pi$ be an irreducible representation of
$GL(n,F)$, Bump and Friedberg associated to $\pi$ an Euler fator
$L(\pi,BF,s_1,s_2)$ in \cite{BF}, that should be equal to
$L(\phi(\pi),s_1)L(\phi(\pi),\Lambda^2,s_2)$, where $\phi(\pi)$ is the
Langlands' parameter of $\pi$. The main result of this paper is to show that
this equality is true when $(s_1,s_2)=(s+1/2,2s)$, for $s$ in $\C$. To prove
this, we classify in terms of distinguished discrete series, generic
representations of $GL(n,F)$ which are $\chi_\alpha$-distinguished by the Levi
subgroup $GL([(n+1)/2],F) \times GL([n/2],F)$, for
$\chi_\alpha(g_1,g_2)=\alpha(det(g_1)/det(g_2))$, where $\alpha$ is a character
of $F^*$ of real part between -1/2 and 1/2. We then adapt the technique of
\cite{CP} to reduce the proof of the equality to the case of discrete series.
The equality for discrete series is a consequence of the relation between
linear periods and Shalika periods for discrete series, and the main result of
\cite{KR}.
