weird bessel zero question

given 'a' and 'b' fixed i define the function

$$f(t)= bJ_{2t}(a)$$

here $J_{n}$ is a Bessel function but in this cases i would be interested in getting the solutions (?? are there any ? ) for

$$J_{2t}(a)=0$$

so for what 'index' 't' is the bessel function evaluated at the point 'a' equal to zero

is there an expansion or a formula asymptotic or whatever for the limit $2t \to \infty$

sorry, as a physicist many of the Bessel function we treated had only integer indices

from the representaion for the Bessel function to high indices is it possible that the solutions are related to the form $t_{n}= \frac{a}{W(bn)}$ for $W(x)$ the Lambert function ??

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1 Answer

Interesting question. Using Maple, I computed $J_{2t}(10)$, we can see some zeros in there.

Here are the positive zeros of $J_{2t}$ for $0 \le t \le 5$

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