| L(s) = 1 | − 3·7-s − 3·11-s + 13-s − 5·17-s − 6·19-s − 3·23-s − 2·29-s + 6·31-s − 3·37-s + 5·41-s + 2·49-s + 7·53-s + 61-s − 15·71-s + 2·73-s + 9·77-s − 15·79-s − 6·83-s + 11·89-s − 3·91-s − 97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 15·119-s + ⋯ |
| L(s) = 1 | − 1.13·7-s − 0.904·11-s + 0.277·13-s − 1.21·17-s − 1.37·19-s − 0.625·23-s − 0.371·29-s + 1.07·31-s − 0.493·37-s + 0.780·41-s + 2/7·49-s + 0.961·53-s + 0.128·61-s − 1.78·71-s + 0.234·73-s + 1.02·77-s − 1.68·79-s − 0.658·83-s + 1.16·89-s − 0.314·91-s − 0.101·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 1.37·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 7 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−14.26874511708280, −13.59624583119553, −13.33414865436410, −12.86963464865278, −12.55162610242610, −11.86445050850099, −11.37727179310182, −10.68984090940748, −10.42148529754246, −9.958617173387835, −9.313064504398946, −8.840370040619090, −8.359645375732294, −7.870334928633851, −7.092787023804426, −6.733226595723907, −6.136831374207989, −5.807854465035933, −5.047144640182708, −4.290792935898633, −4.048452179869079, −3.200643244192047, −2.572196839580656, −2.214282211314659, −1.248080074747580, 0, 0,
1.248080074747580, 2.214282211314659, 2.572196839580656, 3.200643244192047, 4.048452179869079, 4.290792935898633, 5.047144640182708, 5.807854465035933, 6.136831374207989, 6.733226595723907, 7.092787023804426, 7.870334928633851, 8.359645375732294, 8.840370040619090, 9.313064504398946, 9.958617173387835, 10.42148529754246, 10.68984090940748, 11.37727179310182, 11.86445050850099, 12.55162610242610, 12.86963464865278, 13.33414865436410, 13.59624583119553, 14.26874511708280
