In [1] the authors provided a definition and characterization of turning points for the Riemann's zeta function. In this post I denote the Ramanujan's zeta function as $$\varphi(s)=\sum_{n=1}^\infty\frac{\tau(n)}{n^s},$$ where $s$ denotes the complex variable and $\tau(n)$ is the tau function (this function $\varphi(s)$ has an analytic continuation, functional equation, Euler product, and similar properties than the Riemann's zeta function, see the Wolfram MathWorld encyclopedia Tau Dirichlet Series and [2]).
Motivation. I tried to write a characterization of those points $b=\sigma+it$ such that $\Im \varphi(b)=0$ and $\Re\varphi'(b)=0$, compare with the definition that the authors of [1] provided in the section 9. Bound for the real loops. I wrote a similar statement that is showed in my post as Conjecture. But I don't know how to determine the corresponding number $A$, and elaborate more the statement of this conjecture in the spirit of the lemma that provided the authors as a characterization of turning points in the mentioned section of [1] and that I evoke as the following Conjecture.
Conjecture. The point $\sigma+it$ with $\sigma>A$ (I don't know how to set a suitable $A$, I believe that at least is required $A>7$) is a turning point for the function $\varphi(s)$ if and only if $$\Im\left(\sum_p\log\left(1-\frac{\tau(p)}{p^s}+\frac{p^{11}}{p^{2s}}\right)\right)=0\tag{1}$$ and $$\Re\left(\sum_p\frac{(\tau(p)p^s-2p^{11})\log p}{p^{2s}+p^{11}-\tau(p)p^s}\right)=0.\tag{2}$$
Question. A) I would like to know if it possible to elaborate more the expressions $(1)$ and $(2)$ to get a chatacterization of the turning points for the Ramanujan's zeta function $\varphi(s)$ with a better mathematical content* than my expressions $(1)$ and $(2)$. Is it possible to determine or approximate the real number $A$? B) Alternatively, express the turning points of $\varphi(s)$ in other interesting way if you are able. Many thanks.
*That is, if it is posible to improve my expressions $(1)$ or $(2)$ in the spirit of the work that did the authors of [1].
If in the course of your research you've calculated a turning point for $\varphi(s)$ feel free to add it as example.
References:
[1] J. Arias de Reyna and J. van de Lune, Some bounds and limits in the theory of Riemann's zeta function, J. Math. Anal. Appl. 396 (2012) 199-214.
[2] G. H. Hardy, Ramanujan: Twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing (2002).
