L had no reason to suspect you and, honestly, didn't want you involved in the case at all either way. L felt it was best for his personal life to stay out of the case.
Again, L didn't suspect you of being Kira. In fact, you had remained the same person even after Kira started his killings. Your reluctance to move to Japan at first was something L didn't quite understand, but he chalked it up to the issues regarding the recognition of your marriage or other, reasonable reasons. Even, he thought your reluctance to have him investigate Kira was because of your fear of him becoming a target.
But after nearly a year and a half, L started to become suspicious without any promising leads. At first, he had brushed his suspicions off—he was simply being paranoid. You were just a househusband, who cooked and cleaned, waiting for him to return after long days of working. You hadn't given him a reason to doubt your loyalty or intentions.
L started to notice minor oddities in your behavior when he first began spending more time at home. You appeared happy to see him, often preparing his favorite dishes for dinner and dessert, which was normal. But, whenever he asked simple questions (what you had done that day, especially) or discussed any potential breakthroughs in the Kira case, you would become nervous, awkward, and hesitant, sometimes even changing the subject. L noticed that you were particularly uncomfortable when he would lie and describe what they thought Kira to be like similar to you.
L kept his suspicions about you a secret, the others didn’t even know he had a husband anyway. He wanted to investigate you on his own, without your knowledge. He didn't want you to realize that he was looking into your behavior.
As the two of you sat down to enjoy your dinner, you mentioned the name of a criminal who had been killed by Kira earlier that day. L sat up a little straighter, listening intently. The criminal's identity had not yet been disclosed to the public.