Infinite product related to the Wallis product

Some time ago I heard this math question on the radio:

The Wallis product $$\frac{2}{1}*\frac{2}{3}*\frac{4}{3}*\frac{4}{5}*\frac{6}{5}*\frac{6}{7}* \dotsb$$ is known to converge to $\pi/2$, but what does this infinite product converge to: $$\frac{\sqrt[2]{2}}{\sqrt[1]{1}}*\frac{\sqrt[2]{2}}{\sqrt[3]{3}}*\frac{\sqrt[4]{4}}{\sqrt[3]{3}}*\frac{\sqrt[4]{4}}{\sqrt[5]{5}}*\frac{\sqrt[6]{6}}{\sqrt[5]{5}}*\frac{\sqrt[6]{6}}{\sqrt[7]{7}} *\dotsb$$

All I know about it is that it is aproximatly equal to $1.37676673907$
Wolfram Alpha doesn't give me much more

How does one attack such a problem? I know how the value for the Wallis product was found, but none of these techniques seem to work on this seemingly related product.

Tell me anything you find or know about it!

Edit: Thanks to the link lucian gave me I found the closed form: $$2^{2\gamma-\ln{(2)}}$$

I checked 50 digits and they are all correct, now how would you arrive at this or how would you prove this?

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A useful link for these sort of questions would be the Inverse Symbolic Calculator. (The old version can be found here). –  Lucian Mar 4 at 7:42
@Lucian Wow that's really usefull, I found the closed form, I'll edit it in the original post! –  Jens Renders Mar 4 at 10:19