Let $P$ be the prime zeta function

\begin{aligned} {\displaystyle P (s) = \sum _ {p\, \in\mathrm {\, primes}} {\frac {1} {p^{s}}} = {\frac {1} {2^{s}}} + {\frac {1} {3^{s}}} + {\frac {1} {5^{s}}} + {\frac {1} {7^{s}}} + {\frac {1} {11^{s}}} + \cdots } \end{aligned}

and define the semiprime zeta function as

\begin{aligned} & \zeta_{\Omega_ 2} (s) = \exp \sum _k^\infty \frac{P (k s)^2+P(2 k s)}{2 k} \end{aligned}

When these zeros are plugged into

\begin{aligned} &\sum _{\gamma} -\frac{\cos (\gamma \log t)}{\ \sqrt{\gamma}} \end{aligned}

where $\gamma$ is the imaginary part of the zeros of $\zeta_{\Omega_ 2} (\frac{1}{2}+it)$, the sum should peak at the semiprimes.

But the zeros are hard to find. I calculated the near-zeros along the line $1/2 +it$ with

func[t_] := 1/Abs[Exp[(PrimeZetaP[(1/2 + I t)]^2 + PrimeZetaP[2 (1/2 + I t)])/2]];
data = Table[{t, func[t]}, {t, .01, #, .01}] &[(**)10^4];
peaks = Pick[data, PeakDetect[data[[;; , 2]], .01, .001], 1];

in Mathematica, which takes a long time to evaluate, so I uploaded the data here:

peaks = ToExpression[Import["https://raw.githubusercontent.com/martinq321/peaks/main/omega2"]];

Below is a plot of the absolute value of the first term of $\zeta_{\Omega_ 2} (s)$

$\left| \exp \left(P \left (1/2+ i t \right)^2/2+P \left(1+ 2i t \right)/2\right)\right|$ with the red lines marking near-zero points:

Given the above, the plot below shows

\begin{aligned} &\sum _{k \in \text{peaks}} -\frac{\cos (k \log t)}{\ \sqrt{k}}, \quad 200 \geq t \geq 250 \end{aligned}

where, despite the noise, the peaks show at the semiprimes and the primes:

and with the noise cleared up a bit:

li2 = Select[peaks, #[[2]] > 5 &][[All, 1]]; x1 = 200; x2 = 250;
data1 = Table[(Sum[-Cos[li2[[p]] Log[t]]/Sqrt@p, 
{p, Length@li2}]), {t, x1, x2, .1}];
ListLinePlot[BilateralFilter[data1, 1., 1, MaxIterations -> 25], 
PlotStyle -> {AbsoluteThickness[3]}, DataRange -> {x1, x2}, 
ImageSize -> 700, Frame -> True, AspectRatio -> 1/8]

What is going on here? I presume the peaks also showing at the primes because many of the 'zeros' are very close to the zeros of the zeta function. Is there a way to filter them out? Is there a better way to find the zeros of $\zeta_{\Omega_ 2} (s)$?