[hsapipte] Zeta with zeros off the critical line

Multiplying any well-behaved function by (x-a)^2/((x-a)^2+1) inserts a zero at x=a but does not affect the value of the function (very much) elsewhere.

What is a similar transformation that will work on complex valued functions of complex arguments?  The goal here is for fun, to create a function that looks mostly like the Riemann Zeta function, but has a pair of zeros at 0.5 plusminus epsilon + i*t, that can be used for illustrating what a counterexample to the Riemann Hypothesis might look like.

I think this should be easy, though it would be nice if the transformation were elegant, not some ugly piecewise defined function.