I am looking for the references on Taylor series expansion of Riemann xi function at $\frac{1}{2}$.


$$ \xi (s)=\sum\_0^{\infty}a\_{2n}(s-\frac{1}{2})^{2n}$$
where
$$a\_{2n}=4\int\_1^{\infty}\frac{d[x^{3/2}\psi'(x)]}{dx}\frac{(\frac{1}{2}ln(x))^{2n}}{(2n)!}x^{-1/4}dx$$
and
$$\psi(x)=\sum\_{m=1}^{\infty}e^{-m^2\pi x}=\frac{1}{2}[\theta\_3(0,e^{-\pi x})-1]$$


Specifically I would like to know how fast $a\_{2n}$ goes to zero.


Has anyone proved that 
$$a\_0>a\_2>a\_4>...>a\_{2n}>...>a\_{\infty}=0$$ 


Thanks a lot!



References on Taylor series expansion of Riemann xi function
In the paper:


M. W. Coffey, "Asymptotic estimation of $\xi^{(2n)}(1/2)$: On a conjecture
of Farmer and Rhoades", Mathematics of Computation, {\bf 78} (2009) 1147--1154


you may find the first terms of an asymptotic expansion for $\log\xi^{(2n)}(1/2)$.
From it you may get a good estimate of the coefficients $a\_{2n}$.


In particular 


$$\log a\_{2n}=2[1-\log(4n)+\log(\log n)]n-\frac{2n}{\log n}+\frac74\log(2n)-\frac34\log(\log n)+O(1)$$


