There is given Dirichlet series: $$f:\mathbb{C}\rightarrow \mathbb{R}_{\ge 0}$$ $$f(x+iy)=\eta(x+iy)\eta(x-iy)=\sum_{n=1}^{\infty} \frac{a(n)}{n^{x+iy}}$$ Where $a(n)=\sum_{d|n}(-1)^{d+\frac{n}{d}}d^{2yi}$ And of course $x>0$.
Let's consider $g(x,y)=\frac{d^{2}f(x+iy)}{dx^{2}}=\sum_{n=1}^{\infty} \frac{(\ln n)^{2}a(n)}{n^{x+iy}}$.
My question: Is that function always non negative?
I made some research and i deduced that for $x>0$ small enough function $g(x,y)$ is positive hence for constant $y$ if there would be negative value, then it would exist some $x$ that:
$$g(x,y)=0$$
Regards.