  In this paper, we obtain a formula for the special value of Euler-Dirichlet
$L$-function $L_E(s,\chi)$ at $s=1$. This leads to another class number formula
of $\mathbb{Q}(\mu_{m})^{+}$, the maximal real subfield of $m$th cyclotomic
field. From this formula, we construct a new type of cyclotomic units in
$\mathbb{Q}(\mu_{p^{n}})$, which implies a similar Kummer's relation between
the ideal class number of $\mathbb{Q}(\mu_{p^{n}})^{+}$ and the unit index.
