| L(s) = 1 | + 2-s + 3-s − 4-s + 6-s + 7-s − 3·8-s + 9-s − 12-s + 6·13-s + 14-s − 16-s + 18-s + 4·19-s + 21-s − 3·24-s + 6·26-s + 27-s − 28-s + 2·29-s + 5·32-s − 36-s − 6·37-s + 4·38-s + 6·39-s + 10·41-s + 42-s − 4·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.408·6-s + 0.377·7-s − 1.06·8-s + 1/3·9-s − 0.288·12-s + 1.66·13-s + 0.267·14-s − 1/4·16-s + 0.235·18-s + 0.917·19-s + 0.218·21-s − 0.612·24-s + 1.17·26-s + 0.192·27-s − 0.188·28-s + 0.371·29-s + 0.883·32-s − 1/6·36-s − 0.986·37-s + 0.648·38-s + 0.960·39-s + 1.56·41-s + 0.154·42-s − 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 151725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
| 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
−13.60990242729036, −13.23850524055877, −12.72154567049291, −12.36378738368947, −11.57580082115744, −11.41139525497708, −10.76402328136172, −10.16803830633521, −9.646931977423771, −9.192930258601669, −8.691198764272976, −8.248795597410028, −7.983918645725587, −7.135311822330861, −6.652287887780915, −6.089067871924057, −5.442157835267150, −5.218232682249810, −4.410855921758007, −3.943778593567252, −3.594784564109436, −2.955531958259170, −2.457458820564465, −1.395789443581862, −1.114116041270985, 0,
1.114116041270985, 1.395789443581862, 2.457458820564465, 2.955531958259170, 3.594784564109436, 3.943778593567252, 4.410855921758007, 5.218232682249810, 5.442157835267150, 6.089067871924057, 6.652287887780915, 7.135311822330861, 7.983918645725587, 8.248795597410028, 8.691198764272976, 9.192930258601669, 9.646931977423771, 10.16803830633521, 10.76402328136172, 11.41139525497708, 11.57580082115744, 12.36378738368947, 12.72154567049291, 13.23850524055877, 13.60990242729036
