L = Length of unsupported span.
Live loads and dead loads are pounds per square foot so it will remain constant over an area. Load and deflection are not directly proportional - a double 2x6 will not give you twice the span with the same deflection amount. I shouldn't say won't - in some cases it may give you twice the span. What will often happen is the span will increase but deflection will not become an issue - it will start to affect total allowable load.
For instance - 2x6 @ 16" o.c. 8'6" span - 40psf live load, 12psf dead load. The member performs at L/373 for total load.. now double the span and double the joist - you only get L/86.
Now bringing the span down to 10'6" with a double 2x6 the deflection is L/377 but fails due to total load.
These are just quick numbers to show some relationships and there are many more variables to consider than just deflection or loads.. etc.