ellL1

Function: ellL1
Section: elliptic_curves
C-Name: ellL1_bitprec
Prototype: GD0,L,b
Help: ellL1(E, {r = 0}): returns the value at s=1 of the derivative of order r of the L-function of the elliptic curve E.
Doc: returns the value at $s=1$ of the derivative of order $r$ of the
 $L$-function of the elliptic curve $E$.
 \bprog
 ? E = ellinit("11a1"); \\ order of vanishing is 0
 ? ellL1(E)
 %2 = 0.2538418608559106843377589233
 ? E = ellinit("389a1");  \\ order of vanishing is 2
 ? ellL1(E)
 %4 = -5.384067311837218089235032414 E-29
 ? ellL1(E, 1)
 %5 = 0
 ? ellL1(E, 2)
 %6 = 1.518633000576853540460385214
 @eprog\noindent
 The main use of this function, after computing at \emph{low} accuracy the
 order of vanishing using \tet{ellanalyticrank}, is to compute the
 leading term at \emph{high} accuracy to check (or use) the Birch and
 Swinnerton-Dyer conjecture:
 \bprog
 ? \p18
   realprecision = 18 significant digits
 ? E = ellinit("5077a1"); ellanalyticrank(E)
 time = 8 ms.
 %1 = [3, 10.3910994007158041]
 ? \p200
   realprecision = 202 significant digits (200 digits displayed)
 ? ellL1(E, 3)
 time = 104 ms.
 %3 = 10.3910994007158041387518505103609170697263563756570092797@com$[\dots]$
 @eprog