Welcome to the forum, Alex.

Your serial number does seem odd, but then again 'odd' is almost the norm when it comes to the study of serial numbers.

The Model 10 machines that used the 'L' serial number prefix were apparently manufactured between Oct. 1920 and Mar. 1925, and were numbered from 000 to 59,000. Six digit numbers didn't begin until Sept. 1928.

The shape of the front panel and keyboard surround on your machine definitely indicates that your Remington was made after 1920. I think the practice of placing model numbers on the front panel didn't start until 1923, and since your machine also lacks the decimal tabulator, it would have to be a post-1925 model.

There's a 1921 Model 10 in the database with a six digit number, but there aren't any supporting photos, so not much can be gleaned from that machine. 

I'm not a Remington expert, so I can't do much more than take a guess that your machine dates to around 1925. It's possible that the front panel isn't original to the machine. Or perhaps it was rebuilt/refurbished later on when the six digit numbers had begun and given a new serial number at that point. If that's the case then your machine was rebuilt in Oct. 1937.
 

Hello together,

here are some pictures from before and after.

https://www.dropbox.com/sh/k8bqy9vrs7rey3z/AACVJjWcDqi0ToYNgtZ2KOjta?dl=0

Best regards 

Alex