The sub-leading coefficient of the L-function of an elliptic curve

TL;DR

This paper explores the relationship between the leading and sub-leading coefficients of the L-function of an elliptic curve over a number field, providing insights into their interconnected behavior.

Contribution

It establishes a novel relation between the leading and sub-leading coefficients of elliptic curve L-functions, advancing understanding of their analytic properties.

Findings

Identifies a specific relation between the coefficients at s=1

Provides a new perspective on the structure of elliptic curve L-functions

Enhances theoretical understanding of L-function behavior near s=1

Abstract

In this very short note, we show that there is a relation between the leading term at $s=1$ of an $L$-function of an elliptic curve defined over an number field and the term that follows.