The motivation to this question can be found in http://wstein.org/books/bsd/bsd.pdf In page 9 the author claimed that:
1.4.1. Approximating the Rank. Fix an elliptic curve $E$ over $Q$. The usual method to approximate the rank is to find a series that rapidly converges to $L^{(r)}(E, 1)$ for $r = 0, 1, 2, 3, . . .,$ then compute $L(E, 1), L'(E, 1), L''(E, 1)$, etc., until one appears to be nonzero.
My quetion is: How one can determine the series that rapidly converges to $L^{(r)}(E, 1)$ from Theorem 1.10 (Lavrik) in the beginning of the page.