Difficult integral - who can help?

Hello together,

[I posted this already on sci.math, but perhaps you are more familier
with this type of problem. So sorry for cross-posting,. but the problem
is quiet urgent :-( ]
]
During my calculations for a physical problem I derived the following
integral that must be solved:

Integral[z]:=

\int_{-\infinity}^{z} dz' Exp[-i \pi (z'^2 - i \alpha z')] *
\int_{-\infinity}^{z'} dz'' Exp[+i \pi (z''^2 - i \alpha z'')]

As you see its a double-integral.

Please note:

z is real
i = \sqrt(-1)
\alpha is real and also \alpha > 0

I solved the integral for z=\infinity. So I have Integral[\infinity]
which gives 1/2.

But I also need Integral[z]. I asked lots of people and tried some
tricks, but nothing worked for this integral.

Perhaps you can help.

Thanks a lot!!!

Bye,
Mark

.