Can these roots have an equation?

[MisterStarshine]

I know that for example the equation $${e}^{z}-1=0$$ has roots $$z=2\,i\,\pi \,n$$ uniformly spaced along the imaginary axis. Also the equation $$\mathrm{cos}\left( z\right) -1=0$$ has roots $$z=2\,\pi \,n$$ this time along the real line. What if I want a function that repeats itself in a two dimentional pattern? A general way of writing the roots with arbitary starting position a+i*b would be $$z=a+\alpha\,m+i\,\left(b+\beta\,n\right)$$ The variables "m" and "n" here are integers. The variable "alpha" determines the periodicity parallel to the real axis and "beta" for the imaginary, What kind of function would give such roots?