Abstract
Let E be an elliptic curve over , and a prime of good ordinary reduction. The p-adic L-function for always vanishes at s = 1, even though the complex L-function does not have a zero there. The -invariant itself appears on the right-hand side of the formulawhere with . We first devise a method to calculate effectively, then show it is non-trivial for all elliptic curves E of conductor with , and almost all ordinary primes p < 17. Hence, in these cases at least, the order of the zero in at s = 1 is exactly one.
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