Computing L-Invariants for the Symmetric Square of an Elliptic Curve

Let E be an elliptic curve over Q, and p≠2 a prime of good ordinary reduction. The p-adic L-function for Sym²E always vanishes at s = 1, even though the complex L-function does not have a zero there. The L-invariant itself appears on the right-hand side of the formula ddsLp(Sym²E,s)∣∣∣s=1=Lp(Sym²E)×(1−α⁻²ᵖ)(1−pα⁻²ᵖ)×L∞(Sym²E,1)(2πi)⁻¹Ω+EΩ-E where X²−aᵖ(E)X+p=(X−αᵖ)(X−βᵖ) with αp∈Z×ᵖ. We first devise a method to calculate Lp(Sym²E) effectively, then show it is non-trivial for all elliptic curves E of conductor NE≤300 with 4|NE, and almost all ordinary primes p < 17. Hence, in these cases at least, the order of the zero in Lp(Sym²E,s) at s = 1 is exactly one