  We consider the Mellin transforms of certain Chebyshev functions based upon
the Chebyshev polynomials. We show that the transforms have polynomial factors
whose zeros lie all on the critical line or on the real line. The polynomials
with zeros only on the critical line are identified in terms of certain
$_3F_2(1)$ hypergeometric functions. Furthermore, we extend this result to a
1-parameter family of polynomials with zeros only on the critical line. These
polynomials possess the functional equation $p_n(s;\beta)=(-1)^{\lfloor n/2
\rfloor} p_n(1-s;\beta)$. We then present the generalization to the Mellin
transform of certain Gegenbauer functions. The results should be of interest to
special function theory, combinatorics, and analytic number theory.
