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The song (or at least the 1st melody part) goes like this:
d|-------------|-------------|-------------|/15----------| A|-------------|-13----------|/17------13--|-------------| F|---------12--|---------12--|-------------|-------------| C|-------------|-------------|-------------|-------------| G|-------------|-------------|-------------|-------------| D|-------------|-------------|-------------|-------------|
I know what 12/17 means, but not
----|/17 12--|---
You know what 12/17 means: you play the note at the 12th fret and slide to the 17th fret. In this case both notes are important.
But, when you see /17, this means that the first note is really not important. You slide into the note at the 17th fret, but the starting note is not emphasized; this is just an ornament or part of the way that the line is phrased. In practice, usually you would start one or maybe two frets below the target note and just slide right on up to the 17th fret, without lingering at all on the first note; the idea is just to get that sound of sliding into the note to come out.
When you are trying to learn music from tab or even from staff notation, it is not a bad idea to listen to a recording to get help in understanding the phrasing and make sure that you aren't misinterpreting something. If you listened to the recording that this tab goes with, I bet that you could figure out what the notation was trying to tell you.
12/17, as OP has it in the question, refers to a single string. /17 is not connected to the 12 on the other string. If the intention is to slide from 13 to 17, then there would be a 13/17 on the A string, above a 12 on the F string. As it is, /17 just indicates a little pop up from below into the note at 17; this is a very common phrasing device in jazz guitar playing. It could start at 13, or someplace else; it isn't really important where the slide starts.