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