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# Argument selection for T14
Integer arguments belong to the Bernoulli-number, even-zeta and odd-zeta special-value families CITE{SelectionIntegerZeta}. The point $s=1/2$ is fixed by the functional equation $s\mapsto1-s$ CITE{SelectionZetaSymmetry}. The orders $3/2$ and $5/2$ occur in the particle-number and energy integrals of the three-dimensional ideal Bose gas at unit fugacity CITE{SelectionBoseGas}. These reasons do not single out the other nonintegral rational orders of a denominator-bounded mesh.
Coverage: it holds the integer arguments $-40\leq s\leq40$, $s\ne1$, the functional-equation centre $s=1/2$, and the Bose-gas orders $s=3/2,5/2$; the integer cutoff is a finite reference prefix, not a completeness claim beyond that range.
## Primary mathematical evidence
- DLMF 25.6: Integer Arguments: https://dlmf.nist.gov/25.6
- DLMF 25.4: Reflection Formulas: https://dlmf.nist.gov/25.4
- David Tong: Statistical Physics, Chapter 3, Bose gases: https://davidtong.org/pdfs/teaching/statistical-physics/statphys3.pdf
The mathematical deductions in the selection paragraph are distinguished from a source publishing this exact finite list. The range cutoff is a reference limit, not a theorem that every included dimension or order has a separate documented application.
## Reproduction and history
Run `python generate.py` to verify and `python checks.py T14 --live` for independent checks. Neither writes. The source uses exact rational arithmetic and certified Arb balls, with mpmath as a separate agreement check. Retained numeric strings and annotations are copied from, and verified against, the website export; the generator is not claimed to have authored those annotations. Removed sample identities and old attached sources remain in website revision history. No identity is reused for another argument.