| L(s) = 1 | − 3-s − 2·4-s − 5-s + 7-s + 9-s + 6·11-s + 2·12-s + 5·13-s + 15-s + 4·16-s + 2·20-s − 21-s − 23-s + 25-s − 27-s − 2·28-s + 2·29-s + 31-s − 6·33-s − 35-s − 2·36-s + 3·37-s − 5·39-s + 7·41-s − 6·43-s − 12·44-s − 45-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 1.80·11-s + 0.577·12-s + 1.38·13-s + 0.258·15-s + 16-s + 0.447·20-s − 0.218·21-s − 0.208·23-s + 1/5·25-s − 0.192·27-s − 0.377·28-s + 0.371·29-s + 0.179·31-s − 1.04·33-s − 0.169·35-s − 1/3·36-s + 0.493·37-s − 0.800·39-s + 1.09·41-s − 0.914·43-s − 1.80·44-s − 0.149·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30345 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.135516127\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.135516127\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−14.86199932581842, −14.65577164990746, −14.11768287899886, −13.38653098732733, −13.20596358852822, −12.31273671668432, −11.92485859092470, −11.45529944576532, −10.97317057019272, −10.24713192956798, −9.758360856385196, −8.978271981464785, −8.705676182088026, −8.212040799582409, −7.385004812881219, −6.810440884420208, −6.053497585438896, −5.768400521373579, −4.852378365616520, −4.281282468195877, −3.852618758924898, −3.379794937121474, −2.055712903342049, −1.024902289500348, −0.8012350785984288,
0.8012350785984288, 1.024902289500348, 2.055712903342049, 3.379794937121474, 3.852618758924898, 4.281282468195877, 4.852378365616520, 5.768400521373579, 6.053497585438896, 6.810440884420208, 7.385004812881219, 8.212040799582409, 8.705676182088026, 8.978271981464785, 9.758360856385196, 10.24713192956798, 10.97317057019272, 11.45529944576532, 11.92485859092470, 12.31273671668432, 13.20596358852822, 13.38653098732733, 14.11768287899886, 14.65577164990746, 14.86199932581842
