Hasse-Weil L-function ($L(E/F,s)={\prod }_v\det (1-{\mathrm{Frob}}_{v}N{v}^{-s}\mid V_{\ell }(E{)}^{I_v}{)}^{-1}$)

For an elliptic curve over a number field, its Hasse-Weil L-function is the product of the local factors determined by its Tate module. At a good finite place, the factor is $(1-a_{v}N{v}^{-s}+N{v}^{1-2s}{)}^{-1}$, with $a_v=Nv+1-\#E(\kappa (v))$. At a bad place use the inertia-invariant subspace and an auxiliary prime different from the residue characteristic. For a CM elliptic curve with all endomorphisms defined over the base, this factors as the product of the two conjugate Hecke L-functions.