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 nice properties. It's an entire function, unlike the $\Gamma$ and $\zeta$ functions. Its reflection formula $\Xi(-z) = \Xi(z)$ is particularly easy to remember. The Riemann hypothesis for $\Xi(z)$ is also much simpler: all zeros of $\Xi(z)$ are real.
On the other hand, that formula in its definition is a pretty ugly one, and makes it obvious it's pretty much just the zeta function shifted, rotated, and scaled. Is there a nicer representation for it, possibly in terms of an integral or other special functions?