# Is there a closed-form expression for this integral?

Does there exist a closed-form for the integral

$$\int 8\frac { (\zeta(1/2+it))^2\pi}{-\Psi(1,1/4-i/2t) ( \zeta(1/2+it))^2 +\Psi(1,1/4+i/2t) (\zeta(1/2+it))^2 + 8\zeta(2,1/2+it) \zeta(1/2+it) -8(\zeta(1,1/2+it))^2}\,{\rm d}t$$

the latex expression is equivalent to

A graph of the integrand is .. and it has a pole at 5.5611757696135...

I posted a related question at

Does $z (s) = \int_0^s \zeta \left( \frac{1}{2} + i t \right) d t = s + \sum_{n = 2}^{\infty} \frac{i (n^{- i s} - 1)}{\ln (n) \sqrt{n}}$ converge?

• I highly doubt this monstrosity has a closed form. Prove me wrong. – Frank W. Jul 7 '18 at 2:57
• what about just the integral $\int \!\zeta \left( 1/2+it \right) \,{\rm d}t$ then? – crow Jul 7 '18 at 21:49
• I asked a related question at math.stackexchange.com/questions/2845044/… – crow Jul 8 '18 at 22:30
• I thought this might be useful because a root off the line would cause there to be a turning point that doesnt touch zero.. I think – crow Jan 13 at 2:06