# 복소 이차 수체의 데데킨트 제타함수 special values

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## $$s=1$$ 에서의 값

### $q \equiv 3 \pmod{4}$

• $K=\mathbb{Q}(\sqrt{-q})$, $$q \geq 7$$ , $$q \equiv 3 \pmod{4}$$ 인 경우
• $$d_K=-q$$
• $$\chi(a)=\left(\frac{a}{q}\right)$$
• $$\chi(-1)=-1$$, $$\tau(\chi)=i\sqrt{q}$$
• 다음이 성립한다

$L(1,\chi)= \frac{- \pi\sqrt{q}}{q^2}\sum_{a=1}^{q-1}\left(\frac{a}{q}\right) a=\frac{\pi h_K}{\sqrt{q}}$ $h_K=-\sum_{a=1}^{q-1}\left(\frac{a}{q}\right)\frac{a}{q}$

### $q \equiv 1 \pmod{4}$

• $$K=\mathbb{Q}(\sqrt{-q})$$ , $$q \geq 5$$ , $$q \equiv 1 \pmod{4}$$ 인 경우
• $$d_K=-4q$$, $$\chi(-1)=-1$$, $$\tau(\chi)=2i\sqrt{q}$$
• 다음이 성립한다

$L(1,\chi)= -\frac{ \pi\sqrt{q}}{8q^2}\sum_{(a,4q)=1}\chi(a) a=\frac{\pi h_K}{2\sqrt{q}}$ $h_K=-\frac{1}{4}\sum_{(a,4q)=1}\left(\frac{a}{q}\right)\frac{a}{q}$

## $$s=2$$ 에서의 값

• 복소이차수체 $K$에 대하여 다음이 성립한다

$\zeta_{K}(2)=\frac{\pi^2}{6\sqrt{|d_K|}}\sum_{(a,d_k)=1} (\frac{d_K}{a})D(e^{2\pi ia/|d_k|})$

• 예를 들어, $K=\mathbb{Q}\sqrt{-7}$에 대하여, 다음이 성립한다

$\zeta_{\mathbb{Q}\sqrt{-7}}(2)=\frac{\pi^2}{3\sqrt{7}}(D(e^{2\pi i/7})+D(e^{4\pi i/7})-D(e^{6\pi i/7}))$ 여기서 $$D(z)$$는 블로흐-비그너 다이로그(Bloch-Wigner dilogarithm)

$\zeta_{\mathbb{Q}\sqrt{-1}}(2)=1.50670301$ $\zeta_{\mathbb{Q}\sqrt{-2}}(2)=1.75141751\cdots$ $\zeta_{\mathbb{Q}\sqrt{-3}}(2)=\frac{\pi^2}{6\sqrt{3}}(D(e^{2\pi i/3})-D(e^{4\pi i/3}))=\frac{\pi^2}{3\sqrt{3}}D(e^{2\pi i/3})=1.285190955484149\cdots$ $\zeta_{\mathbb{Q}\sqrt{-7}}(2)=\frac{\pi^2}{3\sqrt{7}}(D(e^{2\pi i/7})+D(e^{4\pi i/7})-D(e^{6\pi i/7}))=1.89484145$ $\zeta_{\mathbb{Q}\sqrt{-11}}(2)=1.49613186$

### figure eight knot complement

• $$V=\frac{9\sqrt{3}}{\pi^2}\zeta_{\mathbb{Q}(\sqrt{-3})}(2)=3D(e^{\frac{2i\pi}{3}})=2D(e^{\frac{i\pi}{3}})=2.029883212819\cdots$$
• $$\zeta_{\mathbb{Q}(\sqrt{-3})}(2)=\frac{\pi^2}{3\sqrt{3}}D(e^{\frac{2\pi i}{3}})$$
• $$L_{-3}(2)=\frac{2}{\sqrt{3}}D(e^{\frac{2\pi i}{3}})$$
• 2.02988321281930725

$V(4_{1})=\frac{9\sqrt{3}}{\pi^2}\zeta_{\mathbb{Q}(\sqrt{-3})}(2)=3D(e^{\frac{2i\pi}{3}})=2D(e^{\frac{i\pi}{3}})=2.029883212819\cdots$

## 메모

• $$s=1$$ 에서의 $$L_{d_K}'(1)$$의 값$L_{d_K}'(1)=\frac{2\pi h_K(\gamma+\ln 2\pi)}{w_K \cdot \sqrt{|d_K|}}-\frac{\pi}{\sqrt{|d_K|}}\sum_{(a,d_K)=1}\chi(a)\log\Gamma (\frac{a}{|d_K|})$
• L-함수의 미분 항목 참조