Well, by subtracting $\xi(2s)$ to $E(z,s)$ combined with some growth estimate "seems to be close" to saying that $\xi(2s)y^s$ is part of the constant term of the Fourier series $E(z,s)$. For example, if $s=3/4$ it says that $E(z,s)$ behaves asymptotically exactly like $\xi(3/2)\cdot y^{3/4}$.

Dear Luis, this is a nice characterization. Of course specifying partly what the constant term of the Fourier series is, is not as much conceptual as what I was hoping at first, but may be one cannot do better than that.

Do you have any precise idea on how to fix it?

Thanks Matt, this is good observation! Probably, one should put some explicit restrictions on the constant $C(s)$ which appears implicitly in the big O notation of property (5).

Dear Gunter Harder, I know what is wrong. In the formula appearing in the display if it is $\frac{\partial}{\partial s}E(z,s)$ and therefore this is why you pick up a $\log(y)$. So I think that what I wrote is correct.

Dear Kunnysan, thanks for pointing out the inconsistancy with the functional equation and the shift with the constant!

Dear GH, thanks for the comment you are perfectly. I'll add the Euler factor with the factor $1/2$ so that I at least get the right residues!

Yes indeed, the anecdote behind your first answer is quite inspiring!

OK thanks! This is a very nice argument!

Dear Andras, what do you mean by "take all the posets"?