Questions about the set of values at which a given function evaluates to zero. For questions about "square roots", "cube roots", and such, consider using the (radicals) and (arithmetic) tag. For questions about roots of Lie algebras, use the (lie-algebra) tag instead.

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how to show this function has zeros interlacing and including those of Riemann zeta

Let $\chi (t) = H \left( - \frac{i}{2} (2 t - 1) \right) = \dfrac{4 i \pi \zeta (t) \left( \left( \ddot{\Psi} \left(\frac{t}{2} \right) - \ddot{\Psi} \left( \frac{1}{2} - \frac{t}{2} \right)\right) ...