| L(s) = 1 | − 2-s − 2·3-s + 4-s + 2·6-s + 7-s − 8-s + 9-s − 2·12-s + 4·13-s − 14-s + 16-s + 6·17-s − 18-s − 2·19-s − 2·21-s + 2·24-s − 5·25-s − 4·26-s + 4·27-s + 28-s + 6·29-s + 4·31-s − 32-s − 6·34-s + 36-s + 2·37-s + 2·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.577·12-s + 1.10·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.458·19-s − 0.436·21-s + 0.408·24-s − 25-s − 0.784·26-s + 0.769·27-s + 0.188·28-s + 1.11·29-s + 0.718·31-s − 0.176·32-s − 1.02·34-s + 1/6·36-s + 0.328·37-s + 0.324·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30926 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30926 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 47 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
| 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
−15.49860040194924, −15.02538542340238, −14.32174888009762, −13.76620003730495, −13.25142335299183, −12.32329851420824, −12.11479124157842, −11.49302650001710, −11.20090004313582, −10.45233746418812, −10.17383644872705, −9.616080410246592, −8.725642489789581, −8.239009925136723, −7.935307552601874, −6.976023677008769, −6.525526593178665, −5.950709823788886, −5.472310647259829, −4.870418420918839, −4.022319660034148, −3.325208938740451, −2.494655821274781, −1.440823148497990, −0.9933664688767733, 0,
0.9933664688767733, 1.440823148497990, 2.494655821274781, 3.325208938740451, 4.022319660034148, 4.870418420918839, 5.472310647259829, 5.950709823788886, 6.525526593178665, 6.976023677008769, 7.935307552601874, 8.239009925136723, 8.725642489789581, 9.616080410246592, 10.17383644872705, 10.45233746418812, 11.20090004313582, 11.49302650001710, 12.11479124157842, 12.32329851420824, 13.25142335299183, 13.76620003730495, 14.32174888009762, 15.02538542340238, 15.49860040194924
