The usual way to test for lognormality would be to take logs and test for normality.
Then any suitable test of normality would do; a Shapiro Wilk would be a reasonable choice.
However, this leads to the question "why are you testing that?"
"Check" is not the same as "test". Hypothesis tests are not generally useful as checks of assumptions.
In particular, goodness-of-fit tests are only really useful in a limited set of circumstances -- mostly they're used to answer a question we already know the answer to (and one which is actually not the same one we need an answer to).
Your data aren't lognormal, so there is no point in testing a question you know the answer to. A more relevant question would be 'how badly non-lognormal might it be?' or even better 'how much will this affect my inference?'. Those aren't answered by hypothesis tests. They're more like "effect-size" sorts of problems.