On the Conjecture of Birch and Swinnerton-Dyer for an ... - Up To

AN ELLIPTIC CURVE OF RANK 3 479Taking 250 terms of the series in (12) gives L'(1) = 0 to 13 decimal places. But thisimplies that L'(1)= 0 exactly, since the main result of [2] implies that L'(1) is asimple multiple of the height of some rational point on E ("Heegner point") and, aswe have seen, E contains no rational points of very small nonzero height. Since L(s)has odd order, we have ords= 1 L(s) ? 3.In general, the functions Gr(x) satisfy Go(x) = e-x, Gr'(x) = -(1/x)Gr_1(x), soGr(X) = Pr (log 1-) + E r lrn!for some polynomial Pr of degree r. To determine Pr, we use the integral representation:(13) Gr(X) = 2 fi | () x- ds anyc > 0.(To prove (13), we observe that the right-hand side satisfies the same recursivedifferential equations as Gr(x) and tends to zero as x -> ox.) Shift the path ofintegration in (13) to the left; then the residue at s = -n gives the term (-1)" rx "/nr n!and the residue at s = 0 gives Pr(log 1/x). Hence,Since by Euler-Maclaurinr t "71 ??Pr(t) = Yr-rn m where 17(1 + s) - y,1s"7=0 17 = 0we find, for r = 3, the expansion00(_)log 17(1 + S) = --ys + E ~ (n=2)SnG3 (X)= +~3 -(log? -v)- + (1h1xPG()=6 (lg-Y 12 (lgx 7 17l3 n3n!which converges for all x. Using this we find the valueii=1n(14) lim L( 3s-2 -AG(3~s-(I (s - 1) =n 50771.7318499001193006897919750851using the terms for n < 600 (the error made in breaking off the series here can beestimated using (12) and the formulas G3(x) X-3e-x and _jaj11 d(n)rn, whered(n) is the number of divisors of n).The results of the computations described in this section are summarized in Table2.