| L(s) = 1 | − 9.87·2-s + 9·3-s + 65.5·4-s + 15.4·5-s − 88.8·6-s − 200.·7-s − 330.·8-s + 81·9-s − 152.·10-s − 32.4·11-s + 589.·12-s + 878.·13-s + 1.97e3·14-s + 139.·15-s + 1.17e3·16-s − 446.·17-s − 799.·18-s + 329.·19-s + 1.01e3·20-s − 1.80e3·21-s + 320.·22-s + 839.·23-s − 2.97e3·24-s − 2.88e3·25-s − 8.67e3·26-s + 729·27-s − 1.31e4·28-s + ⋯ |
| L(s) = 1 | − 1.74·2-s + 0.577·3-s + 2.04·4-s + 0.276·5-s − 1.00·6-s − 1.54·7-s − 1.82·8-s + 0.333·9-s − 0.483·10-s − 0.0808·11-s + 1.18·12-s + 1.44·13-s + 2.69·14-s + 0.159·15-s + 1.14·16-s − 0.374·17-s − 0.581·18-s + 0.209·19-s + 0.566·20-s − 0.892·21-s + 0.141·22-s + 0.330·23-s − 1.05·24-s − 0.923·25-s − 2.51·26-s + 0.192·27-s − 3.16·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 9T \) |
| 103 | \( 1 - 1.06e4T \) |
| good | 2 | \( 1 + 9.87T + 32T^{2} \) |
| 5 | \( 1 - 15.4T + 3.12e3T^{2} \) |
| 7 | \( 1 + 200.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 32.4T + 1.61e5T^{2} \) |
| 13 | \( 1 - 878.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 446.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 329.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 839.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.45e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.88e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.60e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.78e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 3.73e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.05e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.03e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.29e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.87e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.65e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.46e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.86e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 9.61e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.68e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 9.77e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.27e4T + 8.58e9T^{2} \) |
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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
−10.16294940105887186538297168081, −9.219993294524627771672243580989, −8.853633018674050460469434131342, −7.73211608152014692295283937351, −6.75321548327538317689083986459, −5.99958994078864601685973857698, −3.68904110059084172560130398996, −2.59534242735737427254218101335, −1.29173534515059188367447794945, 0,
1.29173534515059188367447794945, 2.59534242735737427254218101335, 3.68904110059084172560130398996, 5.99958994078864601685973857698, 6.75321548327538317689083986459, 7.73211608152014692295283937351, 8.853633018674050460469434131342, 9.219993294524627771672243580989, 10.16294940105887186538297168081
