| L(s) = 1 | + 4.79·2-s + 14.9·4-s + 6.25·5-s + 7·7-s + 33.4·8-s + 29.9·10-s − 11·11-s − 35.7·13-s + 33.5·14-s + 40.4·16-s + 133.·17-s + 161.·19-s + 93.6·20-s − 52.7·22-s + 66.2·23-s − 85.8·25-s − 171.·26-s + 104.·28-s + 208.·29-s − 39.3·31-s − 73.5·32-s + 637.·34-s + 43.7·35-s + 197.·37-s + 774.·38-s + 209.·40-s − 434.·41-s + ⋯ |
| L(s) = 1 | + 1.69·2-s + 1.87·4-s + 0.559·5-s + 0.377·7-s + 1.47·8-s + 0.947·10-s − 0.301·11-s − 0.763·13-s + 0.640·14-s + 0.632·16-s + 1.89·17-s + 1.95·19-s + 1.04·20-s − 0.510·22-s + 0.600·23-s − 0.687·25-s − 1.29·26-s + 0.707·28-s + 1.33·29-s − 0.228·31-s − 0.406·32-s + 3.21·34-s + 0.211·35-s + 0.877·37-s + 3.30·38-s + 0.826·40-s − 1.65·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.784038514\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.784038514\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 - 4.79T + 8T^{2} \) |
| 5 | \( 1 - 6.25T + 125T^{2} \) |
| 13 | \( 1 + 35.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 133.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 161.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 66.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 208.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 39.3T + 2.97e4T^{2} \) |
| 37 | \( 1 - 197.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 434.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 375.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 503.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 44.8T + 1.48e5T^{2} \) |
| 59 | \( 1 + 582.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 73.2T + 2.26e5T^{2} \) |
| 67 | \( 1 + 928.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 755.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 277.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 651.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 282.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.04e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.11e3T + 9.12e5T^{2} \) |
| 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
−10.14257513538614044281169621044, −9.509688724506428169826359248300, −7.927201322560204273781636084615, −7.25382942322055545933398159569, −6.10846308542284062292702327809, −5.31179857460304737958220479914, −4.84052083323305641998157362567, −3.44277687262179239301347908663, −2.73220088095683067244704688146, −1.34660957046689371538856528282,
1.34660957046689371538856528282, 2.73220088095683067244704688146, 3.44277687262179239301347908663, 4.84052083323305641998157362567, 5.31179857460304737958220479914, 6.10846308542284062292702327809, 7.25382942322055545933398159569, 7.927201322560204273781636084615, 9.509688724506428169826359248300, 10.14257513538614044281169621044
