What is the nature of illness which makes you want to leave a prestigious company like L &T.
What is the medical documentary proof of illness?
That can convince the company to break their own rules of notice period.
Please show the medical papers and seek early release.
This is very unfortunate situation. Your assessment appears premature and leading you to take hasty decision.
If your decision is unchangeable, quit by giving required notice or persuade management to cut short the notice period. But don't abandon the job which will badly reflect on you.
As stated by you have joined L&T a month back, and as per the industry standards / procedures you might be on Probation, and the period of notice during probation would be one month only.
Please check your appointment letter and take proper relief
Notice period rules depend on the organisation. They can be relaxed on a case by case basis. Since you have a health reason, you can request the HR of your company to relieve you without serving the required notice period. To strengthen your case, please submit the necessary medical documents.