  We analyze metrics for how close an entire function of genus one is to being
real rooted. These metrics arise from truncated Hankel matrix positivity-type
conditions built from power series coefficients at each real point.
Specifically, if such a function satisfies our positivity conditions and has
well-spaced zeros, we show that all of its zeros have to (in some explicitly
quantified sense) be far away from the real axis. The obvious interesting
example arises from the Riemann zeta function, where our positivity conditions
yield a family of relaxations of the Riemann hypothesis. One might guess that
as we tighten our relaxation, the zeros of the zeta function must be close to
the critical line. We show that the opposite occurs: any potential complex
zeros are forced to be farther and farther away from the critical line.
