Linear complementary dual (LCD) codes can be used to against side-channel attacks and fault noninvasive attacks. Let $d_{a}(n,6)$ and $d_{l}(n,6)$ be the minimum weights of all binary optimal linear codes and LCD codes with length $n$ and dimension 6, respectively.In this article, we aim to obtain the values of $d_{l}(n,6)$ for $n\geq 51$ by investigating the nonexistence and constructions of LCD codes with given parameters. Suppose that $s \ge 0$ and $0\leq t\leq 62$ are two integers and $n=63s+t$. Using the theories of defining vectors, generalized anti-codes, reduced codes and nested codes, we exactly determine $d_{l}(n,6)$ for $t \notin\{21,22,25,26,33,34,37,38,45,46\}$, while we show that $d_{l}(n,6)\in$$\{d_{a}(n,6)$ $-1,d_{a}(n,6)\}$ for $t\in\{21,22,26,34,37,38,46\}$ and $ d_{l}(n,6)\in$$ \{d_{a}(n,6)-2,$ $d_{a}(n,6)-1\}$ for$t\in{25,33,45\}$.