| L(s) = 1 | + 2.57·3-s + 1.78·5-s − 7-s + 3.64·9-s − 2.72·11-s − 4.27·13-s + 4.60·15-s − 7.16·19-s − 2.57·21-s + 0.504·23-s − 1.80·25-s + 1.66·27-s + 0.0909·29-s + 1.37·31-s − 7.03·33-s − 1.78·35-s − 7.41·37-s − 11.0·39-s + 7.77·41-s + 7.22·43-s + 6.51·45-s − 11.5·47-s + 49-s + 6.58·53-s − 4.87·55-s − 18.4·57-s + 10.3·59-s + ⋯ |
| L(s) = 1 | + 1.48·3-s + 0.798·5-s − 0.377·7-s + 1.21·9-s − 0.822·11-s − 1.18·13-s + 1.18·15-s − 1.64·19-s − 0.562·21-s + 0.105·23-s − 0.361·25-s + 0.321·27-s + 0.0168·29-s + 0.246·31-s − 1.22·33-s − 0.301·35-s − 1.21·37-s − 1.76·39-s + 1.21·41-s + 1.10·43-s + 0.971·45-s − 1.68·47-s + 0.142·49-s + 0.905·53-s − 0.657·55-s − 2.44·57-s + 1.35·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8092 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8092 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - 2.57T + 3T^{2} \) |
| 5 | \( 1 - 1.78T + 5T^{2} \) |
| 11 | \( 1 + 2.72T + 11T^{2} \) |
| 13 | \( 1 + 4.27T + 13T^{2} \) |
| 19 | \( 1 + 7.16T + 19T^{2} \) |
| 23 | \( 1 - 0.504T + 23T^{2} \) |
| 29 | \( 1 - 0.0909T + 29T^{2} \) |
| 31 | \( 1 - 1.37T + 31T^{2} \) |
| 37 | \( 1 + 7.41T + 37T^{2} \) |
| 41 | \( 1 - 7.77T + 41T^{2} \) |
| 43 | \( 1 - 7.22T + 43T^{2} \) |
| 47 | \( 1 + 11.5T + 47T^{2} \) |
| 53 | \( 1 - 6.58T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 + 3.85T + 61T^{2} \) |
| 67 | \( 1 + 11.4T + 67T^{2} \) |
| 71 | \( 1 - 6.83T + 71T^{2} \) |
| 73 | \( 1 + 1.78T + 73T^{2} \) |
| 79 | \( 1 + 2.06T + 79T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 + 5.09T + 89T^{2} \) |
| 97 | \( 1 + 15.5T + 97T^{2} \) |
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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
−7.53782388794930756840579930987, −6.99706678117726468020079179271, −6.12905054716902502491675265049, −5.39700996550059592537083221774, −4.51346191740198700791219782902, −3.80270119653177311999356152733, −2.74579338832109062944254440274, −2.46235389313294719428892493196, −1.70940383333883724169970554272, 0,
1.70940383333883724169970554272, 2.46235389313294719428892493196, 2.74579338832109062944254440274, 3.80270119653177311999356152733, 4.51346191740198700791219782902, 5.39700996550059592537083221774, 6.12905054716902502491675265049, 6.99706678117726468020079179271, 7.53782388794930756840579930987
