This is an identity. mpmath gives the same value for the hypergeometric function and the polynomial with your values.

@wendt I an still learning something new every day.

Kummer proved it in 1836 in his paper "Uber die hypergeometrische Reihe ..." Journal de Crelle 15 (1836) 127--172 ( & 27). It is not found in the paper of Gauss. Later Riemann obtained it in other way.

@ მამუკა ჯიბლაძე @paul garrett No, I mean $\sigma=1$, I speak of the function $\arg \zeta(1+i t)$. This is real analytic vanishes at $c T$ points (think on the x-ray, but this can be proved ). Also we know how to compute the probabilities I speak about.

In what sense use you the word regular? "... the argument of zeta of the line $\sigma=1$ becomes very regular". Of course it is a real analytic function. But, for $0<t<T$ there are $cT$ points where it is $=0$. But it is not bounded. The probability to be greater than $2\pi$ being very small, to be $\ge 4\pi$ almost incredibly small, and so on. I will not call this behavior "regular".

@user47958 I computed each term with 500 digits. After some experiments with the terms I saw this more than sufficient. Then added the terms one by one. The code is very simple but too long for a comment. Send me an email if you want the code.

with mpmath I compute the sum. It is equal to 0.0000000013060480365934635751811931267930306865307928780

But the last integral is not $-\pi^3/12$. I showed that its value is $0$. Look at my answer in mathoverflow.net/questions/200880/interesting-triple-integral

Please not. TeX is perfect as it is. The change of format of mathematical papers is undesirable.