The first part of this note shows that the odd period polynomial of eachHecke cusp eigenform for full modular group produces via Rodriguez--Villegastransform ([Ro--V]) a polynomial satisfying the functional equation of zetatype and having nontrivial zeros only on the middle line of its critical strip.The second part discusses Chebyshev lambda--structure of the polynomial ring asBorger's descent data to $\bold{F}_1$ and suggests its role in possiblerelation of $\Gamma_{\bold{R}}$--factor to "real geometry over $\bold{F}_1$"(cf. also [CoCons2]).
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