Plot of a Bessel function if possible

Question

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) $

Answer 1

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

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

Answer 2

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

But here is your zeta