Hi @jclmntn,
sorry for asking a stupid question, but are you sure that you are using the same function in both cases?
If the white lines in your Wolfram Alpha plot are considered to have function value zero, I would also expect any function value v(r = 0, q) to be zero. However:
> v(r = 0, q = 0)
[1] 0
> v(r = 0, q = 1)
[1] 8
> v(r = 0, q = -1)
[1] 0
> v(r = 0, q = -0.5)
[1] 0
> v(r = 0, q = -2)
[1] 0
> v(r = 0, q = -3)
[1] 0
> v(r = 0, q = -5)
[1] 0
> v(r = 0, q = +2)
[1] 45
Furthermore, your value at (-1, -1) looks different from zero (red), but
> v(r = -1, q = -1)
[1] 0