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We prove that the first two coefficients in the series expansion around $s=1$of the $p$-adic $L$-function of an elliptic curve over $\mathbb{Q}$ are relatedby a formula involving the conductor of the curve. This is analogous to arecent result of Wuthrich for the classical $L$-function, which makes use ofthe functional equation. We present a few other consequences for the $p$-adic$L$-function and a generalisation to the base-change to an abelian numberfield.

Consequences of the functional equation of the $p$-adic $L$-function of an elliptic curve