I have been told that the Public Policy Institute of California is going to release a poll that supposedly shows a majority against the death penalty. As I have noted many times on this blog, you get dramatically different results depending on how you ask the question. See this post.
Sure enough, PPIC used the skewed-to-the-max wording that implies the respondent is being asked to specify a single punishment for all first-degree murders. If that were the question, I would say LWOP myself. But it is not.

Even if you have issues with the question the trend is still indicative. In 2000 it was 49 for death 47 for LWOP. Now it is 62 for LWOP 31 for death. That certainly represents a downward shift in support for the death penalty.