why there is no value to L3?
L3 (or at least the loop part) is printed on the PCB (as indicated on the schematic) hence isn't a lumped component you need to provide. The antenna is shown (somewhat vaguely) on the PCB diagram in section 15 of the datasheet you took the schematic from.
It would have been helpful if you had provided a direct link to the ...
You don't choose a value for L3. It is a specifically shaped trace on your PCB.
The circuit you gave is a "typical application," but isn't a full design.
Later in the datasheet, the manufacturer shows the circuit and layout for a demo board they produce using the SYN115.
Here's the circuit:
Here's the layout:
L5 is the antenna in the schematic, and L5 ...
The black arrow just means that the antenna/coil has an adjustable tap, that allows it's performance to be adjusted. It shorts out part of the antenna, thus changing it's electrical length.
EDIT 1 - Added IEEE symbol for adjustable inductor
From IEEE Std 315, here's the symbol for an adjustable inductor: