On the other hand, what I saw here:

http://en.wikipedia.org/wiki/Mellin_inversion_theorem

suggests that the Mellin inverse transform requires only the right boundedness in vertical strips, not on a whole half-plane. So it should work...

... but then I tried a numerical test to compare the value of against the inverse Mellin transform of using the Kneser tetrational function as obtained from sheldonison's excellent Kneser-method code (which satisfies the required boundedness criteria). The approximation to the inverse Mellin transform is done by numerically integrating

with some large real value (here, I chose 15) and is any number in (here, I chose 0.5). The following graph shows and on the positive reals from near-0 to 20:

As you can see, decays toward 0, but does not. It seems that for small , but the approximation gets worse at larger values of .

This suggests that there is an error in your derivation of from the inverse Mellin transform, since you should be able to Mellin-transform it back, which means your should decay to 0 along the real axis, but it doesn't. I suspect a formal disproof could be had if you can show that . Whatever your is, it seems most assuredly not to be the inverse Mellin transform of , at least not for the Kneser tetrational (and considering it doesn't seem to provide a convergent Mellin transform, perhaps not the inverse Mellin of anything).

http://en.wikipedia.org/wiki/Mellin_inversion_theorem

suggests that the Mellin inverse transform requires only the right boundedness in vertical strips, not on a whole half-plane. So it should work...

... but then I tried a numerical test to compare the value of against the inverse Mellin transform of using the Kneser tetrational function as obtained from sheldonison's excellent Kneser-method code (which satisfies the required boundedness criteria). The approximation to the inverse Mellin transform is done by numerically integrating

with some large real value (here, I chose 15) and is any number in (here, I chose 0.5). The following graph shows and on the positive reals from near-0 to 20:

As you can see, decays toward 0, but does not. It seems that for small , but the approximation gets worse at larger values of .

This suggests that there is an error in your derivation of from the inverse Mellin transform, since you should be able to Mellin-transform it back, which means your should decay to 0 along the real axis, but it doesn't. I suspect a formal disproof could be had if you can show that . Whatever your is, it seems most assuredly not to be the inverse Mellin transform of , at least not for the Kneser tetrational (and considering it doesn't seem to provide a convergent Mellin transform, perhaps not the inverse Mellin of anything).