If the group of rational points of $E$, which is finitely generated by the Mordell-Weil Theorem, has $g$ generators of infinite order, then the Birch-Swinnerton-Dyer conjecture gives


$L\_E(s)$ has a zero of order $g$ at $s=1$.


Assuming the BSD conjecture, is it possible to (and if so how) to construct such $L\_E(s)$? Specifically, if we want $g=3$ or $4$?



BSD conjecture and L functions with zeroes of order g
I'm not entirely sure what you mean by your question. Here are two remarks:


1. If you *assume* BSD, then to "construct" $L\_E$ you just need to give the curve $E$. There are (many) elliptic curves /$\mathbb{Q}$ whose ranks have been computed, and are (say) equal to 3 or 4.
2. If one wants an example without assuming BSD, then you are in trouble - for given $E$, you can compute $L^{(n)}(s)$ to any desired degree of accuracy, but proving that it vanishes computationally is impossible.


However, two things help you. If the sign in the functional equation is -1, then you have that the order of vanishing is also odd. The Gross-Zagier formula can be used to check the vanishing of the first derivative. For example, this is used in the following paper to exhibit an elliptic curve $E$ whose $L\_E$ provably vanishes to order 3.


On the Conjecture of Birch and Swinnerton-Dyer for an Elliptic Curve of Rank 3 
Author(s): Joe P. Buhler, Benedict H. Gross, Don B. Zagier 
Source: Mathematics of Computation, Vol. 44, No. 170 (Apr., 1985), pp. 473-481


