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I am wondering can you do this with pencil and paper?
you'll need either at least two colors, or some skill with grayscale sketching [and possibly multiple pencil guages]
riemann zeta is a complex function, so any point on the 2d plane of your sketch, needs to get multiple dimensions of information drawn... this is in total four dimensions, and can't be captured accurately without encoding an additional dimension somehow. one typical method that is quite intuitive once you're familiar with it is hue [red/green/blue] to represent complex phase [e.g. red is positive, cyan is negative, green and blue are the imaginary cube roots of 1], and then brightness or saturation to represent magnitude.
geek out a bit on youtube about complex functions [e.g. gamma is another common one] and you'll see tons of examples. they can be drawn manually, it might even boost your intuition much more effectively than passively watching videos... it's just a lot of delicate work and computers do it better.
the work done by 3blue1brown on visualizing Riemann zeta function on the complex plane is beautiful. Take a look at the Manim library.
https://github.com/3b1b/manim
I am assuming this can’t be done on an X-Y plane?
it can be rendered on a x-yi plane
I think I'll build a Riemann zeta function visualizer. Coming soon!