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I know you say "moving beyond zeta function jokes", but I'd say the following two zeta-regularizations deserve to be alongside your Ramanujan example: $$\infty!= \sqrt{2\pi}\qquad\qquad\mbox{and }\qquad\qquad \prod_{\mbox{p prime}}p =4\pi^2.$$ One can also entertain beginning calculus students with $\frac{1}{2}!=\frac{1}{2}\sqrt{\pi}$ as a way of introducing the Gamma function.

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Anything involving zeta-regularization would probably fit

I know you say "moving beyond zeta function jokes", but I'd say the bill following two zeta-regularizations deserve to be alongside your Ramanujan example. The two examples that spring to mind are : $$\infty!= \sqrt{2\pi}$$ and $$\prod_{\mbox{p sqrt{2\pi}\qquad\qquad\mbox{and }\qquad\qquad \prod_{\mbox{p prime}}p =4\pi^2.$$

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