| L(s) = 1 | − 3-s + 2·5-s + 9-s + 6·13-s − 2·15-s − 17-s + 8·23-s − 25-s − 27-s − 6·29-s − 8·31-s + 10·37-s − 6·39-s + 6·41-s − 12·43-s + 2·45-s + 51-s − 10·53-s − 8·59-s − 6·61-s + 12·65-s − 12·67-s − 8·69-s + 6·73-s + 75-s + 8·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1/3·9-s + 1.66·13-s − 0.516·15-s − 0.242·17-s + 1.66·23-s − 1/5·25-s − 0.192·27-s − 1.11·29-s − 1.43·31-s + 1.64·37-s − 0.960·39-s + 0.937·41-s − 1.82·43-s + 0.298·45-s + 0.140·51-s − 1.37·53-s − 1.04·59-s − 0.768·61-s + 1.48·65-s − 1.46·67-s − 0.963·69-s + 0.702·73-s + 0.115·75-s + 0.900·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.05321446187649, −14.60930529477696, −13.81078965575082, −13.41992216027841, −12.97065873710832, −12.72076592488550, −11.75245090680044, −11.27700149848067, −10.84346580705203, −10.56606359827229, −9.656736406251172, −9.146506335105158, −9.013524877742196, −7.972356152627851, −7.582701399159548, −6.738631377994659, −6.217039481478766, −5.934280437774619, −5.212748399956355, −4.733054392187898, −3.828771656301505, −3.354992931069669, −2.457674062743786, −1.580237168073311, −1.189180276005915, 0,
1.189180276005915, 1.580237168073311, 2.457674062743786, 3.354992931069669, 3.828771656301505, 4.733054392187898, 5.212748399956355, 5.934280437774619, 6.217039481478766, 6.738631377994659, 7.582701399159548, 7.972356152627851, 9.013524877742196, 9.146506335105158, 9.656736406251172, 10.56606359827229, 10.84346580705203, 11.27700149848067, 11.75245090680044, 12.72076592488550, 12.97065873710832, 13.41992216027841, 13.81078965575082, 14.60930529477696, 15.05321446187649
