| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 4·7-s − 8-s + 9-s − 10-s + 12-s + 2·13-s + 4·14-s + 15-s + 16-s + 2·17-s − 18-s + 19-s + 20-s − 4·21-s − 8·23-s − 24-s + 25-s − 2·26-s + 27-s − 4·28-s − 6·29-s − 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s + 0.554·13-s + 1.06·14-s + 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.229·19-s + 0.223·20-s − 0.872·21-s − 1.66·23-s − 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s − 0.755·28-s − 1.11·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68970 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68970 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.43751582988310, −13.79503541893938, −13.35938712098156, −13.10090496209292, −12.29292177089829, −12.00752355455667, −11.40737969540063, −10.54493295610084, −10.08738318125064, −9.969400497393526, −9.353229854750989, −8.798817931374785, −8.469436613218283, −7.701470250106712, −7.301240854331516, −6.551664379285023, −6.276737922982503, −5.690608348154023, −5.027140022761562, −3.997332044091373, −3.448991357477112, −3.181236038738672, −2.199199826608085, −1.839575914655458, −0.8456319312248338, 0,
0.8456319312248338, 1.839575914655458, 2.199199826608085, 3.181236038738672, 3.448991357477112, 3.997332044091373, 5.027140022761562, 5.690608348154023, 6.276737922982503, 6.551664379285023, 7.301240854331516, 7.701470250106712, 8.469436613218283, 8.798817931374785, 9.353229854750989, 9.969400497393526, 10.08738318125064, 10.54493295610084, 11.40737969540063, 12.00752355455667, 12.29292177089829, 13.10090496209292, 13.35938712098156, 13.79503541893938, 14.43751582988310
