Definition ($L(E,s)$). Define

$$L(E,s)=\prod _p{L}_p(E,s)$$

where

$$L_p(E,s)=\{\begin{array}{cc}{(1-a_p{p}^{-s}+p^{1-2s})}^{-1}\hfill & \text{if }\text{good reduction}\hfill \\ {(1-p^{-s})}^{-1}\hfill & \text{if split multiplicative reduction}\hfill \\ {(1+p^{-s})}^{-1}\hfill & \text{if nonsplit multiplicative reduction}\hfill \\ 1\hfill & \text{if additive reduction}\hfill \end{array}$$

where $\#Ẽ({𝔽}_p)=1+p-a_p$.