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