| L(s) = 1 | − 0.688·2-s − 3-s − 1.52·4-s − 0.688·5-s + 0.688·6-s + 1.90·7-s + 2.42·8-s + 9-s + 0.474·10-s − 11-s + 1.52·12-s − 0.836·13-s − 1.31·14-s + 0.688·15-s + 1.37·16-s − 17-s − 0.688·18-s + 0.428·19-s + 1.05·20-s − 1.90·21-s + 0.688·22-s − 3.05·23-s − 2.42·24-s − 4.52·25-s + 0.576·26-s − 27-s − 2.90·28-s + ⋯ |
| L(s) = 1 | − 0.487·2-s − 0.577·3-s − 0.762·4-s − 0.308·5-s + 0.281·6-s + 0.719·7-s + 0.858·8-s + 0.333·9-s + 0.150·10-s − 0.301·11-s + 0.440·12-s − 0.232·13-s − 0.350·14-s + 0.177·15-s + 0.344·16-s − 0.242·17-s − 0.162·18-s + 0.0983·19-s + 0.234·20-s − 0.415·21-s + 0.146·22-s − 0.636·23-s − 0.495·24-s − 0.905·25-s + 0.113·26-s − 0.192·27-s − 0.548·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 561 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 561 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 + 0.688T + 2T^{2} \) |
| 5 | \( 1 + 0.688T + 5T^{2} \) |
| 7 | \( 1 - 1.90T + 7T^{2} \) |
| 13 | \( 1 + 0.836T + 13T^{2} \) |
| 19 | \( 1 - 0.428T + 19T^{2} \) |
| 23 | \( 1 + 3.05T + 23T^{2} \) |
| 29 | \( 1 + 6.26T + 29T^{2} \) |
| 31 | \( 1 - 2.42T + 31T^{2} \) |
| 37 | \( 1 - 3.59T + 37T^{2} \) |
| 41 | \( 1 + 8.87T + 41T^{2} \) |
| 43 | \( 1 + 11.8T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 - 3.36T + 53T^{2} \) |
| 59 | \( 1 + 0.357T + 59T^{2} \) |
| 61 | \( 1 - 8.14T + 61T^{2} \) |
| 67 | \( 1 - 14.5T + 67T^{2} \) |
| 71 | \( 1 + 5.93T + 71T^{2} \) |
| 73 | \( 1 - 9.14T + 73T^{2} \) |
| 79 | \( 1 + 7.95T + 79T^{2} \) |
| 83 | \( 1 + 8.79T + 83T^{2} \) |
| 89 | \( 1 + 1.88T + 89T^{2} \) |
| 97 | \( 1 + 2.96T + 97T^{2} \) |
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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
−10.14656820119457732359846993019, −9.634176817005075175488913385584, −8.358133029561340345562504699319, −7.935944439946082907884611218015, −6.82376986239609105798281223511, −5.48053854577489052075551319013, −4.75684100565982553567845412211, −3.73886908370954969353921574404, −1.72811843848698989769297322206, 0,
1.72811843848698989769297322206, 3.73886908370954969353921574404, 4.75684100565982553567845412211, 5.48053854577489052075551319013, 6.82376986239609105798281223511, 7.935944439946082907884611218015, 8.358133029561340345562504699319, 9.634176817005075175488913385584, 10.14656820119457732359846993019
