| L(s) = 1 | − 0.710·3-s + 1.82·5-s − 2.49·9-s − 5.77·11-s + 5.36·13-s − 1.29·15-s + 4.21·17-s + 3.23·19-s − 23-s − 1.67·25-s + 3.90·27-s − 5.40·29-s − 1.66·31-s + 4.10·33-s + 9.55·37-s − 3.81·39-s − 9.89·41-s − 7.54·43-s − 4.55·45-s + 1.83·47-s − 2.99·51-s − 10.8·53-s − 10.5·55-s − 2.30·57-s − 6.38·59-s + 9.11·61-s + 9.78·65-s + ⋯ |
| L(s) = 1 | − 0.410·3-s + 0.815·5-s − 0.831·9-s − 1.73·11-s + 1.48·13-s − 0.334·15-s + 1.02·17-s + 0.742·19-s − 0.208·23-s − 0.334·25-s + 0.751·27-s − 1.00·29-s − 0.299·31-s + 0.713·33-s + 1.57·37-s − 0.610·39-s − 1.54·41-s − 1.15·43-s − 0.678·45-s + 0.268·47-s − 0.418·51-s − 1.48·53-s − 1.41·55-s − 0.304·57-s − 0.831·59-s + 1.16·61-s + 1.21·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 0.710T + 3T^{2} \) |
| 5 | \( 1 - 1.82T + 5T^{2} \) |
| 11 | \( 1 + 5.77T + 11T^{2} \) |
| 13 | \( 1 - 5.36T + 13T^{2} \) |
| 17 | \( 1 - 4.21T + 17T^{2} \) |
| 19 | \( 1 - 3.23T + 19T^{2} \) |
| 29 | \( 1 + 5.40T + 29T^{2} \) |
| 31 | \( 1 + 1.66T + 31T^{2} \) |
| 37 | \( 1 - 9.55T + 37T^{2} \) |
| 41 | \( 1 + 9.89T + 41T^{2} \) |
| 43 | \( 1 + 7.54T + 43T^{2} \) |
| 47 | \( 1 - 1.83T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 + 6.38T + 59T^{2} \) |
| 61 | \( 1 - 9.11T + 61T^{2} \) |
| 67 | \( 1 + 10.9T + 67T^{2} \) |
| 71 | \( 1 - 0.315T + 71T^{2} \) |
| 73 | \( 1 - 14.7T + 73T^{2} \) |
| 79 | \( 1 - 10.8T + 79T^{2} \) |
| 83 | \( 1 - 8.54T + 83T^{2} \) |
| 89 | \( 1 + 17.3T + 89T^{2} \) |
| 97 | \( 1 - 8.01T + 97T^{2} \) |
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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
−7.58088332262062798789046301569, −6.40577010984442965091541288595, −5.96830118385178529012881898171, −5.36514280982922486228637994871, −5.00435510288034313696941773742, −3.64225765516869543791078068760, −3.08378018058241282274098991538, −2.18895202645756136193313541915, −1.22846903404109240393717190136, 0,
1.22846903404109240393717190136, 2.18895202645756136193313541915, 3.08378018058241282274098991538, 3.64225765516869543791078068760, 5.00435510288034313696941773742, 5.36514280982922486228637994871, 5.96830118385178529012881898171, 6.40577010984442965091541288595, 7.58088332262062798789046301569
