| L(s) = 1 | − 2·2-s + 2·4-s − 4·5-s − 3·7-s + 8·10-s − 6·11-s − 6·13-s + 6·14-s − 4·16-s − 6·17-s − 2·19-s − 8·20-s + 12·22-s − 4·23-s + 11·25-s + 12·26-s − 6·28-s − 5·29-s + 8·32-s + 12·34-s + 12·35-s − 10·37-s + 4·38-s − 11·41-s − 4·43-s − 12·44-s + 8·46-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 1.78·5-s − 1.13·7-s + 2.52·10-s − 1.80·11-s − 1.66·13-s + 1.60·14-s − 16-s − 1.45·17-s − 0.458·19-s − 1.78·20-s + 2.55·22-s − 0.834·23-s + 11/5·25-s + 2.35·26-s − 1.13·28-s − 0.928·29-s + 1.41·32-s + 2.05·34-s + 2.02·35-s − 1.64·37-s + 0.648·38-s − 1.71·41-s − 0.609·43-s − 1.80·44-s + 1.17·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27747 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27747 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3083 | \( 1 - T \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
| 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
−16.09997515405076, −15.71931847924568, −15.18674678732021, −14.90788891179648, −13.80143775769919, −13.15073353961886, −12.80325808225579, −12.12940277837635, −11.69136238587864, −10.96845321885564, −10.57974916223648, −10.01095406811885, −9.634300884407691, −8.736223055903312, −8.454860338216507, −7.790108792145772, −7.442332841281900, −6.919305939461962, −6.432182031867998, −5.083773324489366, −4.832868793164920, −3.968411038936890, −3.221769264638103, −2.548522070614842, −1.849011659675712, 0, 0, 0,
1.849011659675712, 2.548522070614842, 3.221769264638103, 3.968411038936890, 4.832868793164920, 5.083773324489366, 6.432182031867998, 6.919305939461962, 7.442332841281900, 7.790108792145772, 8.454860338216507, 8.736223055903312, 9.634300884407691, 10.01095406811885, 10.57974916223648, 10.96845321885564, 11.69136238587864, 12.12940277837635, 12.80325808225579, 13.15073353961886, 13.80143775769919, 14.90788891179648, 15.18674678732021, 15.71931847924568, 16.09997515405076
