L(s) = 1 | − 1.08·2-s − 30.8·4-s − 25·5-s − 139.·7-s + 67.9·8-s + 27.0·10-s + 121·11-s − 646.·13-s + 150.·14-s + 913.·16-s + 1.37e3·17-s + 1.90e3·19-s + 770.·20-s − 130.·22-s − 343.·23-s + 625·25-s + 698.·26-s + 4.30e3·28-s − 53.5·29-s + 634.·31-s − 3.16e3·32-s − 1.49e3·34-s + 3.48e3·35-s + 1.16e4·37-s − 2.06e3·38-s − 1.69e3·40-s − 1.88e4·41-s + ⋯ |
L(s) = 1 | − 0.191·2-s − 0.963·4-s − 0.447·5-s − 1.07·7-s + 0.375·8-s + 0.0854·10-s + 0.301·11-s − 1.06·13-s + 0.205·14-s + 0.891·16-s + 1.15·17-s + 1.21·19-s + 0.430·20-s − 0.0576·22-s − 0.135·23-s + 0.200·25-s + 0.202·26-s + 1.03·28-s − 0.0118·29-s + 0.118·31-s − 0.545·32-s − 0.221·34-s + 0.481·35-s + 1.40·37-s − 0.231·38-s − 0.167·40-s − 1.75·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + 25T \) |
| 11 | \( 1 - 121T \) |
good | 2 | \( 1 + 1.08T + 32T^{2} \) |
| 7 | \( 1 + 139.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 646.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.37e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.90e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 343.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 53.5T + 2.05e7T^{2} \) |
| 31 | \( 1 - 634.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.16e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.88e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.33e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.25e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 8.90e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.12e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.18e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.73e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.10e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.60e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 883.T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.39e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.17e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.58e5T + 8.58e9T^{2} \) |
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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
−9.786933515172844498396175787468, −8.970359519836336292430568783602, −7.86838214065113783636151243362, −7.17678415380728784425510634181, −5.87178833177433471847123245910, −4.91520514685328850814074401651, −3.80870189604633262763517229755, −2.94257371112952454596774972605, −1.03121892672350698630691836067, 0,
1.03121892672350698630691836067, 2.94257371112952454596774972605, 3.80870189604633262763517229755, 4.91520514685328850814074401651, 5.87178833177433471847123245910, 7.17678415380728784425510634181, 7.86838214065113783636151243362, 8.970359519836336292430568783602, 9.786933515172844498396175787468