The Riemann $ Xi $ function, defined as

$$

Xi (z) equiv – frac {1} {2} left (z ^ 2 + frac {1} {4} right) pi ^ { frac {1} {4} + i frac {z} {2}} Gamma left ( frac {1} {4} + i frac {z} {2} right) zeta left ( frac {1} {2} + iz right )

$$

has a number of beautiful properties. It's an entire function, unlike the $ Gamma $ and $ zeta $ the functions. His reflection formula $ Xi (-z) = Xi (z) $ is particularly easy to remember. The Riemann hypothesis for $ Xi (z) $ is also much simpler: all the zeros of $ Xi (z) $ are real.

On the other hand, this formula, in its definition, is very ugly and it is obvious that it is about all that the zeta function is shifted, rotated and scaled. Is there a better representation, possibly in the form of an integral function or other?