Zeros of functions on the complex plane

zetafunction
what is the relationship (if any) of the following statement

- A function has ALL the zeros on the line (complex plane) $$Re (z) = A$$ for some Real A

- A function has ALL the zeros on the unit circle defined by $$|z| \le 1$$

i think there is a transformation of coordinates so the line Re (z) = A would become the unit circle but i do not know what is.