  In math.NT/9907019 we proposed an analog of the classical Riemann hypothesis
for characteristic p valued L-series based on the work of Wan, Diaz-Vargas,
Thakur, Poonen, and Sheats for the zeta function $\zeta_{\Fr[\theta]}(s)$.
During the writing of math.NT/9907019, we made two assumptions that have
subsequently proved to be incorrect. The first assumption is that we can ignore
the trivial zeroes of characteristic p L-series in formulating our conjectures.
Instead, we show here how the trivial zeroes influence nearby zeroes and so
lead to counter-examples of the original Riemann hypothesis analog. We then
sketch an approach to handling such ``near-trivial'' zeroes via Hensel's and
Krasner's Lemmas (whereas classically one uses Gamma-factors). Moreover, we
show that $\zeta_{\Fr[\theta]}(s)$ is not representative of general L-series
as, surprisingly, all its zeroes are near-trivial, much as the Artin-Weil
zeta-function of $\mathbb{P}^1/\Fr$ is not representative of general complex
L-functions of curves over finite fields. Consequently, the ``critical zeroes''
(= all zeroes not effected by the trivial zeroes) of characteristic p L-series
now appear to be quite mysterious. The second assumption made while writing
math.NT/9907019 is that certain Taylor expansions of classical L-series of
number fields would exhibit complicated behavior with respect to their zeroes.
We present a simple argument that this is not so, and, at the same time, give a
characterization of functional equations.
