functional equation for the Riemann Xi function

The Riemann Xi Function satisfies the following functional equation:

$$\mathrm{\Xi }(s)=\mathrm{\Xi }(1-s)$$

This equation directly implies the Riemann Zeta function’s functional equation.

This equation plays an important role in the theory of the Riemann Zeta function. It allows one to analytically continue the Zeta and the Xi functions to the whole complex plane. The definition of the Zeta function as a series is only valid when $\mathrm{\Re }(s)>1$. By using this equation, one can express the values of these two functions when in terms of the values when $\mathrm{\Re }(s)>1$. As an illustration of its importance, one can cite the fact that there are no zeros of the Zeta function with real part greater than 1, so without this functional equation the study of the Zeta function would be very limited.