Unfortunately your test is invalid because (according to your recent schematic) your Vcc supply to the 74LS241 is only 3.5 V. This is outside of the allowed range for a normal 74LS device (normal 74LS power supply voltage is 5 V +/- 5% i.e. 4.75 V to 5.25 V). Therefore any results you get using a power supply of 3.5 V are not guaranteed to be those described in the datasheet.
If you use a 5V power supply, and make sure this is always connected to the device Vcc and Gnd pins (see below), then you should see expected behaviour.
And when I connect the Vcc pin [...]
So your first tests were done without any power connected to the Vcc pin! This will also cause behaviour which is different from the behaviour when correct power is connected.
I've done quite a lot of research since first writing this answer, and with unpowered 74LS devices and an input voltage between 0 V and 5.5 V, you may have "got away" without causing damage doing this - just resulting in a high impedance output, which would fit with the behaviour you described. You said:
I have put a led on the 3 output pin, in order to test the output. Then i wired the 19 and 17 pin with 3.5v, but my led isn't showing any high input.
Which I have interpreted as being:
You have [connected an LED + resistor between pin 3 and Gnd], in order to [view the state of] the output. Then [you connected 3.5 V to input pins 17 + 19, with nothing connected to Vcc pin 20] but [your LED didn't light despite the logic] high input[s].
This is not unexpected, since you didn't power the IC via its power pins for this test. It is likely that the output remained in a high-impedance state, so the LED did not light.
Then you said:
when I connect the Vcc pin, without 19 and 17, my led is on, where I expected this to be off.
Which I have interpreted as:
When you connected [3.5 V, as shown in your schematic, to] the Vcc pin, without [any connection to pins] 19 and 17, [your LED] is on, where [you] expected this to be off.
The LED being lit is not a surprise. Although it is not "best practice" to do this, unconnected LS TTL inputs will (usually) float high, as @PeterBennet has already kindly explained.
Therefore, although 3.5 V is outside of the power supply specification, we can expect input pins 17 and 19 to float high and behave as if they were connected to Vcc. Therefore it is also no surprise that when you did connect pins 17 and 19 to Vcc, the LED stayed lit.
Texas Instruments discusses some of these results of reduced voltage power supply voltages to ICs, in this short "Designing With Logic" document.
Unless you know exactly what you are doing and its consequences, do not apply voltages to the other pins of an IC, without the correct power supply voltage(s) also being supplied to its specified power pins (i.e. correctly connecting Gnd / Vss pins, as well as the Vcc / Vdd pins).
If you want to see more examples (related to modern CMOS devices rather than LS TTL) here are some other EE.SE topics where powering devices via other pins, instead of the power pins, caused unexpected behaviour: