For any
$\mathfrak p$ in $\Val(F)-S$, we consider the local term
of the $L$-function associated to the geometric Picard group
which is defined by
\[L_{\mathfrak p}(s,\Pic(\Vbar))=
\frac{1}{\Det(1-(\cardinal\Ffp)^{-s}\Fr_{\mathfrak p}\mid
\Pic(\calli V_{\overline{\Ffp}})\otimes_\ZZ\QQ)}\]
where $

\begin{equation}
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\begin{gathered}
\begin{picture}(240,80)(0,10)
\p(0,90){L(0)}\p(50,90){L'(0)}\p(100,90){L''(0)}
\p(0,60){L(1)}\p(50,60){L'(1)}\p(100,60){L''(1)}
\p(0,30){L(2)}\p(50,30){L'(2)}\p(100,30){L''(2)}
\put(0,45){\vector(0,1){8}}
\put(0,45){\vector(0,-1){8}}
\put(75,45){\vector(1,1){10}}
\put(75,45){\vector(1,-1){10}}
\put(75,45){\vector(-1,1){10}}
\put(75,45){\vector(-1,-1){10}}
\p(150,90){L(0)}\p(200,90){L'(0)}\p(250,90){L''(0)}
\p(150,60){L(1)}\p(200,60){L'(1)}\p(250,60){L''(1)}
\p(150,30){L(2)}\p(200,30){L'(2)}\p(250,30){L''(2)}
\end{picture}\\
\begin{picture}(100,90)(0,10)
\p(0,90){L(0)}\p(50,90){L'(0)}\p(100,90){L''(0)}
\p(0,60){L(1)}\p(50,60){L'(1)}\p(100,60){L''(1)}
\p(0,30){L(2)}\p(50,30){L'(2)}\p(100,30){L''(2)}
\end{picture}
\end{gathered}
\end{equation}