| L(s) = 1 | − 3-s + 9-s − 3·11-s − 13-s + 7·19-s − 5·23-s − 27-s − 6·31-s + 3·33-s − 3·37-s + 39-s − 3·41-s + 8·43-s − 47-s − 5·53-s − 7·57-s + 4·59-s − 8·61-s + 5·69-s + 6·71-s + 14·73-s + 16·79-s + 81-s − 16·83-s + 6·89-s + 6·93-s − 16·97-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 0.904·11-s − 0.277·13-s + 1.60·19-s − 1.04·23-s − 0.192·27-s − 1.07·31-s + 0.522·33-s − 0.493·37-s + 0.160·39-s − 0.468·41-s + 1.21·43-s − 0.145·47-s − 0.686·53-s − 0.927·57-s + 0.520·59-s − 1.02·61-s + 0.601·69-s + 0.712·71-s + 1.63·73-s + 1.80·79-s + 1/9·81-s − 1.75·83-s + 0.635·89-s + 0.622·93-s − 1.62·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−14.48001063564703, −14.02336617045600, −13.63573309469435, −13.01594243429628, −12.38207963234553, −12.22073453613578, −11.48865563481841, −11.04253908627065, −10.58438437422572, −9.929726994037472, −9.620923822531094, −9.013681583439209, −8.264938579622215, −7.637651311541489, −7.448506053626501, −6.684480368680603, −6.083648974973753, −5.384280474926454, −5.227169326379796, −4.475528336293003, −3.724070329929866, −3.182024623839162, −2.373372695579773, −1.722895187441416, −0.8135058003352899, 0,
0.8135058003352899, 1.722895187441416, 2.373372695579773, 3.182024623839162, 3.724070329929866, 4.475528336293003, 5.227169326379796, 5.384280474926454, 6.083648974973753, 6.684480368680603, 7.448506053626501, 7.637651311541489, 8.264938579622215, 9.013681583439209, 9.620923822531094, 9.929726994037472, 10.58438437422572, 11.04253908627065, 11.48865563481841, 12.22073453613578, 12.38207963234553, 13.01594243429628, 13.63573309469435, 14.02336617045600, 14.48001063564703
