Seeking a proven explicit or closed form upper bound on the infinite product of the Zeta function over the natural numbers, 2 to infinity

Question

I'm interested in upper bounds on the product $\prod_{n=2}^\infty \zeta(n)$

The following post was very helpful and from the answers I see I'll need to study up on partitions to learn more about this particular infinite product:

Infinite product of Zetafunctions

Understanding that the actual limit of the product may never be expressible in closed form, for now I'm just hoping to learn if there is ANY proven closed-form upper bound. Say for example could someone have shown that $\prod_{n=2}^\infty \zeta(n)<e$, hypothetically.

Or even a rational number to as many digits as possible. If anyone has provided any values in a paper. I'm really just looking for a nailed down explicit value to the greatest accuracy known so far.

Answer 1

Yes, the log of your product is the sum of $\log\zeta_n$. Each term is less than $\zeta_n-1$ so, by the other post, the log of your product is less than one.

Answer 2

$$\prod_{n=2}^\infty \zeta(n)=2.294856591673313794183\cdots$$ $$\sum _{n=2}^\infty \log (\zeta (n))=0.830670354271780112508\cdots$$