# Plot of a Bessel function if possible

i would like to know where i could find a plot of

$$J_{ia}(2\pi i)$$ (1)

using Quantum mechanics i have conjectured that if $a= \frac{x}{2}$ and $i= \sqrt{-1}$ then

$$J_{it}(2\pi i)\approx0=\zeta (1/2+2it)$$ at least for big $t \to \infty$ (2)

however i do not know how to check or disprove this fact.

the idea is that the Operator $$-D^{2}y(x)+4\pi ^{2}e^{4x}y(x)=E_{n}y(x)$$ (3)

has a Weyl term for the Eigenvalues as $N(T)= \frac{\sqrt{T}}{2\pi}log( \frac{\sqrt{T}}{2\pi e})$

inthe same fashion as the Riemann zeta function

the condition (2) is stablished by imposing that the eigenvalue problem satisfy $y(0)=0=y(\infty)$

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## 2 Answers

Here is what I get in Maple (first real part, then imaginary part)

But here is your zeta

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Here is a plot over some sample values in Wolfram Alpha:

http://www.wolframalpha.com/share/clip?f=d41d8cd98f00b204e9800998ecf8427emmfihklhut

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it is strange i believed it was real... how about $$J_{ait}(2\pi i)+ J_{-ait}(2\pi i)$$ –  Jose Garcia Jan 2 '13 at 15:22
Try to plot $J_{ait}(2\pi)$ which is an even function of $t$. –  mike Sep 8 at 5:07