| L(s) = 1 | − 0.182·3-s + 1.30·5-s − 4.37·7-s − 2.96·9-s + 6.13·11-s − 13-s − 0.239·15-s + 4.69·17-s − 4.05·19-s + 0.800·21-s + 4.43·23-s − 3.28·25-s + 1.09·27-s + 29-s − 5.70·31-s − 1.12·33-s − 5.72·35-s + 6.51·37-s + 0.182·39-s − 9.77·41-s − 3.97·43-s − 3.88·45-s + 6.34·47-s + 12.1·49-s − 0.858·51-s − 5.48·53-s + 8.02·55-s + ⋯ |
| L(s) = 1 | − 0.105·3-s + 0.585·5-s − 1.65·7-s − 0.988·9-s + 1.84·11-s − 0.277·13-s − 0.0617·15-s + 1.13·17-s − 0.930·19-s + 0.174·21-s + 0.925·23-s − 0.657·25-s + 0.209·27-s + 0.185·29-s − 1.02·31-s − 0.195·33-s − 0.968·35-s + 1.07·37-s + 0.0292·39-s − 1.52·41-s − 0.605·43-s − 0.578·45-s + 0.925·47-s + 1.73·49-s − 0.120·51-s − 0.753·53-s + 1.08·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + 0.182T + 3T^{2} \) |
| 5 | \( 1 - 1.30T + 5T^{2} \) |
| 7 | \( 1 + 4.37T + 7T^{2} \) |
| 11 | \( 1 - 6.13T + 11T^{2} \) |
| 17 | \( 1 - 4.69T + 17T^{2} \) |
| 19 | \( 1 + 4.05T + 19T^{2} \) |
| 23 | \( 1 - 4.43T + 23T^{2} \) |
| 31 | \( 1 + 5.70T + 31T^{2} \) |
| 37 | \( 1 - 6.51T + 37T^{2} \) |
| 41 | \( 1 + 9.77T + 41T^{2} \) |
| 43 | \( 1 + 3.97T + 43T^{2} \) |
| 47 | \( 1 - 6.34T + 47T^{2} \) |
| 53 | \( 1 + 5.48T + 53T^{2} \) |
| 59 | \( 1 + 0.0899T + 59T^{2} \) |
| 61 | \( 1 + 10.2T + 61T^{2} \) |
| 67 | \( 1 - 4.34T + 67T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 - 7.05T + 73T^{2} \) |
| 79 | \( 1 - 5.03T + 79T^{2} \) |
| 83 | \( 1 + 2.74T + 83T^{2} \) |
| 89 | \( 1 - 15.5T + 89T^{2} \) |
| 97 | \( 1 + 3.20T + 97T^{2} \) |
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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
−7.61548022942423865598464277534, −6.76505918441725819852246191511, −6.23986056348088361951206812898, −5.89360230099156861515831414270, −4.92728318354999144721605180933, −3.75007657523347241129985192726, −3.35195640387247625119807524727, −2.42076025337729275018954247861, −1.26781981486895434380105025148, 0,
1.26781981486895434380105025148, 2.42076025337729275018954247861, 3.35195640387247625119807524727, 3.75007657523347241129985192726, 4.92728318354999144721605180933, 5.89360230099156861515831414270, 6.23986056348088361951206812898, 6.76505918441725819852246191511, 7.61548022942423865598464277534
