Is it significant that token length of source is close to e?

What's the relationship between Shannon entropy of the distribution, and token length? For example, English is quoted as having a token length of ~4 characters, but source code (that I've tested) seems to be closer to 2.7. Is this significant that it's close to e (i.e., the base of the natural logarithm)? Is source code a more efficient and natural representation of structure/knowledge/information than English? Any thoughts? Any connection with how log appears in thermodynamic entropy?