| L(s) = 1 | − 2-s − 3-s + 4-s + 3.43·5-s + 6-s − 0.177·7-s − 8-s + 9-s − 3.43·10-s − 4.21·11-s − 12-s + 6.72·13-s + 0.177·14-s − 3.43·15-s + 16-s − 17-s − 18-s − 6.85·19-s + 3.43·20-s + 0.177·21-s + 4.21·22-s + 3.46·23-s + 24-s + 6.78·25-s − 6.72·26-s − 27-s − 0.177·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.53·5-s + 0.408·6-s − 0.0671·7-s − 0.353·8-s + 0.333·9-s − 1.08·10-s − 1.27·11-s − 0.288·12-s + 1.86·13-s + 0.0474·14-s − 0.886·15-s + 0.250·16-s − 0.242·17-s − 0.235·18-s − 1.57·19-s + 0.767·20-s + 0.0387·21-s + 0.899·22-s + 0.722·23-s + 0.204·24-s + 1.35·25-s − 1.31·26-s − 0.192·27-s − 0.0335·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 5 | \( 1 - 3.43T + 5T^{2} \) |
| 7 | \( 1 + 0.177T + 7T^{2} \) |
| 11 | \( 1 + 4.21T + 11T^{2} \) |
| 13 | \( 1 - 6.72T + 13T^{2} \) |
| 19 | \( 1 + 6.85T + 19T^{2} \) |
| 23 | \( 1 - 3.46T + 23T^{2} \) |
| 29 | \( 1 + 4.28T + 29T^{2} \) |
| 31 | \( 1 + 8.07T + 31T^{2} \) |
| 37 | \( 1 + 8.31T + 37T^{2} \) |
| 41 | \( 1 - 4.55T + 41T^{2} \) |
| 43 | \( 1 - 7.07T + 43T^{2} \) |
| 47 | \( 1 - 3.46T + 47T^{2} \) |
| 53 | \( 1 + 1.92T + 53T^{2} \) |
| 61 | \( 1 + 11.3T + 61T^{2} \) |
| 67 | \( 1 - 2.33T + 67T^{2} \) |
| 71 | \( 1 + 6.54T + 71T^{2} \) |
| 73 | \( 1 - 1.67T + 73T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 + 5.52T + 83T^{2} \) |
| 89 | \( 1 + 3.84T + 89T^{2} \) |
| 97 | \( 1 + 4.92T + 97T^{2} \) |
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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
−7.72980734581671197859613606457, −6.92629961881000393863587220407, −6.17010689537185387238212151001, −5.81456359059512593028218078167, −5.17516194221685032627480136518, −4.06008970013805355528192243815, −2.95619305299139042830611124525, −2.01174788080385031461140839344, −1.40297907660403678351419025737, 0,
1.40297907660403678351419025737, 2.01174788080385031461140839344, 2.95619305299139042830611124525, 4.06008970013805355528192243815, 5.17516194221685032627480136518, 5.81456359059512593028218078167, 6.17010689537185387238212151001, 6.92629961881000393863587220407, 7.72980734581671197859613606457
