| L(s) = 1 | − 2-s − 3.41·3-s + 4-s + 0.0151·5-s + 3.41·6-s − 2.13·7-s − 8-s + 8.67·9-s − 0.0151·10-s + 4.39·11-s − 3.41·12-s − 4.28·13-s + 2.13·14-s − 0.0516·15-s + 16-s − 0.201·17-s − 8.67·18-s − 3.00·19-s + 0.0151·20-s + 7.28·21-s − 4.39·22-s + 5.10·23-s + 3.41·24-s − 4.99·25-s + 4.28·26-s − 19.3·27-s − 2.13·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.97·3-s + 0.5·4-s + 0.00675·5-s + 1.39·6-s − 0.805·7-s − 0.353·8-s + 2.89·9-s − 0.00477·10-s + 1.32·11-s − 0.986·12-s − 1.18·13-s + 0.569·14-s − 0.0133·15-s + 0.250·16-s − 0.0489·17-s − 2.04·18-s − 0.688·19-s + 0.00337·20-s + 1.59·21-s − 0.936·22-s + 1.06·23-s + 0.697·24-s − 0.999·25-s + 0.840·26-s − 3.73·27-s − 0.402·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3011 | \( 1+O(T) \) |
| good | 3 | \( 1 + 3.41T + 3T^{2} \) |
| 5 | \( 1 - 0.0151T + 5T^{2} \) |
| 7 | \( 1 + 2.13T + 7T^{2} \) |
| 11 | \( 1 - 4.39T + 11T^{2} \) |
| 13 | \( 1 + 4.28T + 13T^{2} \) |
| 17 | \( 1 + 0.201T + 17T^{2} \) |
| 19 | \( 1 + 3.00T + 19T^{2} \) |
| 23 | \( 1 - 5.10T + 23T^{2} \) |
| 29 | \( 1 - 2.44T + 29T^{2} \) |
| 31 | \( 1 - 7.95T + 31T^{2} \) |
| 37 | \( 1 - 9.17T + 37T^{2} \) |
| 41 | \( 1 + 8.64T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 - 9.64T + 47T^{2} \) |
| 53 | \( 1 + 8.68T + 53T^{2} \) |
| 59 | \( 1 + 0.684T + 59T^{2} \) |
| 61 | \( 1 + 2.90T + 61T^{2} \) |
| 67 | \( 1 + 1.50T + 67T^{2} \) |
| 71 | \( 1 - 8.92T + 71T^{2} \) |
| 73 | \( 1 + 14.0T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 - 9.74T + 89T^{2} \) |
| 97 | \( 1 + 12.7T + 97T^{2} \) |
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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
−7.34687705111494823846166788699, −6.93684196972463829103668107641, −6.26232055306949376377373507660, −5.97646702842745963743344180703, −4.80743223444460098106403450780, −4.40016344002551575932906498951, −3.23442297193002481397487037687, −1.88869151841517835104658208817, −0.918996218892914722141104541677, 0,
0.918996218892914722141104541677, 1.88869151841517835104658208817, 3.23442297193002481397487037687, 4.40016344002551575932906498951, 4.80743223444460098106403450780, 5.97646702842745963743344180703, 6.26232055306949376377373507660, 6.93684196972463829103668107641, 7.34687705111494823846166788699
