A principle around the Ramanujan's zeta function in short intervals

Question

Here $s=\sigma+it$ denotes the complex variable. We denote the Ramanujan's zeta function $$\varphi(s)=\sum_{n=1}^\infty\frac{\tau(n)}{n^s}$$

for $\Re(s)>7$, where $\tau(n)$ is the Ramanujan tau function.

While I was studying a video by Harper, a video from YouTube with title The Riemann zeta function in short intervals - Harper - Bourbaki - 30/03/19 from the official channel Institut Henri Poincaré, I wondered if a similar principle than the professor shows as Principle I (see also PRINCIPLE 1.3 from [2]) works in some suitable sense for the Ramanujan's zeta function $\varphi(s)$.

Principle. For any $s$ with $\Re(s)\geq \frac{11}{6}$ (or at least $\Re(s)> \frac{11}{6}$) and $|\Im(s)|\geq 1$ we have $$\varphi(s)\cdot\prod_{\substack{p\text{ prime }\\p\leq X(s)}}\left(1-\frac{\tau(p)}{p^s}+\frac{p^{11}}{p^{2s}}\right)\simeq 1\tag{1}$$ in some "suitable sense".

Question. Does previous Principle involving the identity $(1)$ work for any suitable sense? Explain your words. Many thanks.

Thus I am asking about the explanation of the meaning of previous Principle involving the identity $(1)$. I just wrote it as a similar statement than the Principle I from Harper's colloquium (at few first minutes of the colloquium), but I have no knowledges to know the meaning of it.

References:

[1] G. H. Hardy, Ramanujan: Twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing (2002).

[2] Adam J. Harper, The Riemann zeta function in short intervals [after Najnudel, and Arguin, Belius, Bourgade, Radziwiłł, and Soundararajan], Séminaire BOURBAKI, 71e année, 2018–2019, no 1159 (Mars 2019).

Answer 1

It seems, essentially, that you want to estimate the $L$-function

$L(s,\Delta)=\sum_{n=1}^{\infty}\tau(n) n^{-s}=\prod_p (1-\tau(p)p^{-s}+p^{11-2s})^{-1}$

as a short Euler product, in the sense that there exists a reasonably small integer $X(s)>0$ such that

$L(s,\Delta)\approx\prod_{p\leq X(s)} (1-\tau(p)p^{-s}+p^{11-2s})^{-1}$

with a reasonably small error. This question has been well-studied for $L$-functions in various families when $s$ lies on the line $\mathrm{Re}(s)=1$, though one can place $s$ elsewhere in the critical strip. The literature focusing on $\mathrm{Re}(s)=1$ is very extensive, so I'll focus on that for now.

Rather than studying this problem for a single $L$-function, it is often a bit more interesting to study the problem for, say, the family of $L$-functions associated to weight $k$ level 1 holomorphic cuspidal newforms, and understand the average and limiting behavior as $k\to\infty$. Here is an excellent paper on this matter by Lau and Wu: https://hal.archives-ouvertes.fr/hal-00097046/document. One could also fix the weight, say $k=2$, and pursue similar questions as the level $N$ varies. Here is an excellent paper on this matter by Cogdell and Michel: http://www.math.osu.edu/~cogdell.1/moments-www.ps. Both of these papers were inspired in part by the work of Granville and Soundararajan, where they considered the same problem for Dirichlet $L$-functions associated to real characters: https://arxiv.org/abs/math/0206031.

ADDED: I was mixing normalizations. For the normalization in the original question, $\mathrm{Re}(s)=1$ should be changed to $\mathrm{Re}(s)=13/2$