Take Gauss' two-dimensional map of imaginary numbers. This map charts the numbers that we shall feed into the zeta function. We can lay this map out flat on the table. What we want to do is to create a physical landscape above this map. The height above each imaginary number x should record the answer once this imaginary number is fed into the zeta function.

The trouble is that this answer _(x) also has two coordinates, defining a
point in a second map of the imaginary world. The height of the landscape above the point in the first map can only record one number. So we are going to have to lose some information, just as our two-dimensional shadow cannot tell us everything about our three-dimensional bodies.

We have a number of choices for what we could take the height of the
landscape to be above x in the map we have laid out on the table. We could just read off one of the two coordinates of the answer _(x) and forget the other one. But it turns out that the best shadow to take is to calculate the distance that the output is from the origin in the map. Call this distance D. It is a good measure of how big the output is. The height of the 3-dimensional landscape above the point x will therefore be D.

So what does this 3-dimensional shadow of the zeta function look like?