I'd have another suggestion to replace your $Z(t/log(t))$ : there's the Riemann-Siegel Theta function described in Harold Edwards' book, $theta(t)$. The Gram points satisfy: $theta(g_n) = npi$, n = 1, 2, 3, ... So the idea is to look at $W(alpha) = Z(theta^{-1}(alpha))$ . That way, if $g_n$ is the nth Gram point, $theta^{-1}(npi) = g_n$ and
$W(npi) = Z(theta^{-1}(npi)) = Z(g_n) = (-1)^n zeta(1/2 + i g_n)$ .

Cf. `Gram's Law' at MathWorld: <http://mathworld.wolfram.com/GramsLaw.html > .

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 $[npi, (n+1)pi]$ for W(.) ...

I'd have another suggestion to replace your $Z(t/\log(t))$ : there's the Riemann-Siegel Theta function described in Harold Edwards' book, $\theta(t)$. The Gram points satisfy: $\theta(g_n) = n\pi$, $n$ = 1, 2, 3, ... So the idea is to look at $W(\alpha) = Z(\theta^{-1}(\alpha))$ . That way, if $g_n$ is the $n$th Gram point, $\theta^{-1}(n\pi) = g_n$ and
$W(n\pi) = Z(\theta^{-1}(n\pi)) = Z(g_n) = (-1)^n \zeta(1/2 + i g_n)$ .

Cf. `Gram's Law' at MathWorld: <http://mathworld.wolfram.com/GramsLaw.html > .

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 $[n\pi, (n+1)\pi]$ for $W(.)$ ...