sandwich asked that I put a few screens up, so here's what the function looks like:
x is going from 0 to 5 here. Looks a bit like the 1/x function here. Notice the pole at x=1.
The negative portion of the above graph. x varies from -5 to 0, and the y value has been restricted between -0.01 and 0.03 to show the zeros at each of the even negatives.
This is a plot of the function along the line x=1/2 and a variable imaginary portion. (x goes from 0 to 30i and y goes from -1.5 to 1.5) The important zeros can be seen here; note that they appear to be irrational.
This is a 3D plot of z=|
z(x+iy)|, with x going from 1 to 4 and y from 0 to 40. The depressions in the hills correspond to the x=1/2 line; the value of the function goes down to zero in each of those ditch-like areas. The x=1 pole is the black thing in the corner.
It is fairly easy to compute values for all positive even integers and negative odd and even integers - some examples are the following:
z(-2)=0
z(-1)=-1/12
z(0)=-1/2
z(2)=
p^2/6
z(4)=
p^4/90
z(6)=
p^6/945
z(½+i 14.134725200497455...)=0 (this is one of those nontrivial zeros)
Unfortunately, analyzing anything other than these integers (excluding positive odds, which are also hard) is much more difficult.
And yes, I spend all day and night with this stuff.
