That LM317 is an adjustable linear regulator that takes whatever input voltage you have and outputs a particular output voltage, probably something around 2.5v - 5v . That voltage probably goes to the numeric led display which probably has a tiny board under it with a microcontroller that reads the set output voltage (and maybe current) from the switching regulator and displays it on the numeric display.
If the linear regulator isn't working, then that 3 digit numeric display won't turn on, and quite possibly the input led also won't work, but the switching regulator is independent and may still work.
My suggestion would be to look at the pinout of that LM317 ( it should be input voltage , ground/adjust , output voltage, not necessarily in this order, find a datasheet or just use your multimeter's continuity mode to see which pin goes directly to Input +v and which one goes to ground / Input - on that two wire header..
With power going in through those two wires, then measure the output voltage of that linear regulator .. if you don't see anything there's your problem - the regulator's dead. You can also figure out what output voltage the regulator is supposed to do by the two resistors connected to the gnd/adjust pin (see datasheet) .. if you measure something other than what's supposed to be set by resistors, again the regulator's probably dead.
Next point of failure would be the diodes. I think the D1 diode is used as some cheap basic reverse voltage protection - if you wired the board with the input wires the wrong way around and left it there for a bit of time, you may have popped that diode, which means the switching regulator won't see any input voltage.
Check the other diodes as well... use the diode measurement function on your meter.
Next, look at the LM2596 switching regulator ic ... look at the datasheet and look at the pinout. Figure out the input voltage + and gnd pins and put the multimeter on those and if you don't measure anything, there's something faulty (open) between this input pin or gnd pin and the input voltage header.