1Function: ellL1
2Section: elliptic_curves
3C-Name: ellL1_bitprec
4Prototype: GD0,L,b
5Help: 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.
6Doc: returns the value at $s=1$ of the derivative of order $r$ of the
7 $L$-function of the elliptic curve $E$.
8 \bprog
9 ? E = ellinit("11a1"); \\ order of vanishing is 0
10 ? ellL1(E)
11 %2 = 0.2538418608559106843377589233
12 ? E = ellinit("389a1");  \\ order of vanishing is 2
13 ? ellL1(E)
14 %4 = -5.384067311837218089235032414 E-29
15 ? ellL1(E, 1)
16 %5 = 0
17 ? ellL1(E, 2)
18 %6 = 1.518633000576853540460385214
19 @eprog\noindent
20 The main use of this function, after computing at \emph{low} accuracy the
21 order of vanishing using \tet{ellanalyticrank}, is to compute the
22 leading term at \emph{high} accuracy to check (or use) the Birch and
23 Swinnerton-Dyer conjecture:
24 \bprog
25 ? \p18
26   realprecision = 18 significant digits
27 ? E = ellinit("5077a1"); ellanalyticrank(E)
28 time = 8 ms.
29 %1 = [3, 10.3910994007158041]
30 ? \p200
31   realprecision = 202 significant digits (200 digits displayed)
32 ? ellL1(E, 3)
33 time = 104 ms.
34 %3 = 10.3910994007158041387518505103609170697263563756570092797@com$[\dots]$
35 @eprog
36