Hi Irfan. Ah,.. oops!
I did say that I thought the voltage of the zener used here might be wrong as it looks like the original schematic was generated by a computer program. In my long experience (40 years), I have found that all the theory in the world develops some very strange habits when we let it lose in the wild.
I have a looked at this schematic again and as the forward voltage drop on a red LED is approx 1.6 volts and the voltage drop from base to emitter on the transistor will be 0.6 volts, (due to the minute current the resistor is not doing much) then the zener then should be,.. 13.8 – 0.6 – 1.6 = 11.6 volts. Now that is a very odd value and I suggest making the zener 10 volts with the resistor R4 a 5K preset potentiometer, then adjust it till the circuit shuts off at the suggested ‘float charge’ level of 13.6 volts which is standard for all lead-acid bateries, or just a little higher than that. It is possible to charge to a higher voltage but that may also cause the battery to heat up and shorten its working life-span. I’m not saying that the circuit goes into a float state, it shuts off completely at the max value you set.
You also mention that putting your finger on the end of the zener causes the circuit to switch off. Yes it will.
This is because our bodies operate with electrical impules in the nerves that are actually a lot higher than some might think. Add to that the fact that we also pick up any mains interference like an aerial, thus what is happening is that our bodies generate quite enough voltage to interfere with a simple base junction in a transistor! This is the buzzing noise you get if you touch the input lead to an audio amplifier. It is also why sensitive electronic components are packaged in anti-static bags and foam.
If there is a lot of electrical noise in the envoronment (remember there is a mains transformer very close), adding a capacitor of around 100nF from the bottom of the zener to 0 volts should dump that noise and stop EMI (electro magnetic interference) from triggering the transistor at random.
Just to be certain, as I have just had a delivery of 10 x LM317 chips, I will dig out some bits tomorrow and actually put this circuit together myself to pinpoint where the computer that originally designed it, got it so wrong. I will update the schematic accordingly.