All genuine Elgin watches will have two serial numbers, one on the case and one on the movement.
The only one in terms of value that you should look at and write down, is the serial number on the actual movement.
Many people believe that the case is the most important thing and when advertising focus on that.
If you have the original box, bill of sale or associated information then this will always add value.
Then you should consider the condition of the watch.
Ask for two quotes, as you will need an auction or real value quote and a quote for insurance purposes.
The one for insurance purposes will typically have a higher value as it should cover the real cost of replacement.
The more jewels the movement is set on then the smoother the watch will run and as such generally speaking the more valuable it will be.