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11352 bytes, as of the version from 2026-11-09 18:56 (current). Recorded here, not run.
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"text": "(2)\t\n\u039b\n\ud835\udc41\n(\n\ud835\udc53\n,\n\ud835\udc60\n)\n=\n(\n\ud835\udc41\n2\n\ud835\udf0b\n)\n\ud835\udc60\n\u0393\n(\n\ud835\udc60\n)\n\ud835\udc3f\n(\n\ud835\udc53\n,\n\ud835\udc60\n)\n,\n\u039b\n\ud835\udc41\n(\n\ud835\udc53\n,\n\ud835\udc60\n)\n=\n\ud835\udf00\n\ud835\udc53\n\u039b\n\ud835\udc41\n(\n\ud835\udc53\n,\n\ud835\udc58\n\u2212\n\ud835\udc60\n)\n,\n\ud835\udf00\n\ud835\udc53\n=\n(\n\u2212\n1\n)\n\ud835\udc58\n/\n2\n\ud835\udc64\n\ud835\udc41\n(\n\ud835\udc53\n)\n.\n \n\ud835\udc64\n\ud835\udc41\n(\n\ud835\udc53\n)\n\u2208\n{\n\u2212\n1\n,\n1\n}\n is the eigenvalue of the normalized Fricke involution. The roster records this exact sign [3].",
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"text": "(3)\t\n\ud835\udc5f\n\ud835\udc53\n(\n\ud835\udc4b\n)\n=\n\u2212\n\ud835\udf00\n\ud835\udc53\ni\n\ud835\udc58\n(\n\ud835\udc41\n\ud835\udc4b\n)\n\ud835\udc58\n\u2212\n2\n\ud835\udc5f\n\ud835\udc53\n(\n\u2212\n1\n\ud835\udc41\n\ud835\udc4b\n)\n,\n\ud835\udc5f\n\ud835\udc53\n(\n\u2212\n\ud835\udc4b\n\u2015\n)\n=\n\u2212\n\ud835\udc5f\n\ud835\udc53\n(\n\ud835\udc4b\n)\n\u2015\n.\n The first relation follows by substituting the completed functional equation into (1). The second follows from the reality of the Fourier coefficients. Thus reflection across the imaginary axis preserves the root set.",
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"text": "(2)\t\n\u039b\n\ud835\udc41\n(\n\ud835\udc53\n,\n\ud835\udc60\n)\n=\n(\n\ud835\udc41\n2\n\ud835\udf0b\n)\n\ud835\udc60\n\u0393\n(\n\ud835\udc60\n)\n\ud835\udc3f\n(\n\ud835\udc53\n,\n\ud835\udc60\n)\n,\n \n\u039b\n\ud835\udc41\n(\n\ud835\udc53\n,\n\ud835\udc60\n)\n=\n\ud835\udf00\n\ud835\udc53\n\u039b\n\ud835\udc41\n(\n\ud835\udc53\n,\n\ud835\udc58\n\u2212\n\ud835\udc60\n)\n,\n \n\ud835\udf00\n\ud835\udc53\n=\n(\n\u2212\n1\n)\n\ud835\udc58\n/\n2\n\ud835\udc64\n\ud835\udc41\n(\n\ud835\udc53\n)\n.\n \n\ud835\udc64\n\ud835\udc41\n(\n\ud835\udc53\n)\n\u2208\n{\n\u2212\n1\n,\n1\n}\n is the eigenvalue of the normalized Fricke involution. The roster records this exact sign [3].",
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"locator": "Equations (1.1)-(1.4); completed L-function definition and equation; the reciprocal exponent was derived directly in exact algebra."
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"limitations": "Presentation-only repair. No independent decimal root recomputation was performed in this session; the prior critique is preserved. Exact algebra verifies the identity labels, not the analytic accuracy of stored roots."
}