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QuadraticLValueSquare

Abstract

Two genuine primitive functional equations for a self-dual nonprincipal character eliminate the root-number sign. Nonvanishing at one permits the cancellation. For odd characters the Gamma factors yield L(0)^2=N*(L(1)/pi)^2.

Two genuine primitive functional equations for a self-dual nonprincipal character eliminate the root-number sign. Nonvanishing at one permits the cancellation. For odd characters the Gamma factors yield L(0)^2=N*(L(1)/pi)^2.

Four-slot escape registrations are unfinished, tracked at https://github.com/the-omega-institute/trureturing/issues/26321. No declared_validated status is claimed. This registration debt is separate from the native mathematical proof.

Theorem 1.1 (completed_zero_sq).

Lean statement: D5/S3/Analytic/Zeta/QuadraticLValueSquare.completed_zero_sq

Proof. Machine-checked in Lean as D5/S3/Analytic/Zeta/QuadraticLValueSquare.completed_zero_sq (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Deyi Chen and Zhi-Wei Sun (2024). On determinants involving (j+k/p) plus or minus (j-k/p). DOI: 10.48550/arXiv.2409.08213. URL: https://arxiv.org/html/2409.08213v4#S1.E11.

Commentary.

Two applications of the functional equation eliminate the root number without choosing its sign. This is an equality for the actual completed L-functions.

Theorem 1.2 (zero_sq).

Lean statement: D5/S3/Analytic/Zeta/QuadraticLValueSquare.zero_sq

Proof. Machine-checked in Lean as D5/S3/Analytic/Zeta/QuadraticLValueSquare.zero_sq (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Deyi Chen and Zhi-Wei Sun (2024). On determinants involving (j+k/p) plus or minus (j-k/p). DOI: 10.48550/arXiv.2409.08213. URL: https://arxiv.org/html/2409.08213v4#S1.E11.

Commentary.

For odd characters the two Gamma factors are 1 and the reciprocal of pi. No root-number evaluation is used in this uncompleted L-value identity.

References