I'd have another suggestion to replace your $Z(t/log(t))$ Z(t/\log(t))$ :
there's the Riemann-Siegel Theta function described in
Harold Edwards' book, $theta(t)$. \theta(t)$. The Gram points
satisfy: $theta(g_n) \theta(g_n) = npi$n\pi$, n $n$ = 1, 2, 3, ... So the idea is
to look at $W(alpha) W(\alpha) = Z(theta^{-1}(alpha))$ Z(\theta^{-1}(\alpha))$ .
That way, if $g_n$ is the nth $n$th Gram point,
$theta^{-1}(npi) \theta^{-1}(n\pi) = g_n$ and
$W(npi) W(n\pi) = Z(theta^{-1}(npi)Z(\theta^{-1}(n\pi)) = Z(g_n) = (-1)^n \zeta(1/2 + i g_n)$ .
Cf. `Gram's Law' at MathWorld: .
Perhaps there's a way to rescale $W(.)$ vertically from $Z(.)$ to get identical square-integrals over corresponding intervals say $[g_n, g_{n+1}]$ for Z and $[npi, [n\pi, (n+1)pi]$ n+1)\pi]$ for W(.) $W(.)$ ...
