| L(s) = 1 | − 1.84·2-s − 1.69·3-s + 1.39·4-s + 3.12·6-s + 1.54·7-s + 1.11·8-s − 0.120·9-s + 2.53·11-s − 2.36·12-s + 6.36·13-s − 2.83·14-s − 4.84·16-s − 3.54·17-s + 0.222·18-s + 0.713·19-s − 2.61·21-s − 4.67·22-s + 1.66·23-s − 1.89·24-s − 11.7·26-s + 5.29·27-s + 2.14·28-s − 8.23·29-s + 10.5·31-s + 6.69·32-s − 4.30·33-s + 6.52·34-s + ⋯ |
| L(s) = 1 | − 1.30·2-s − 0.979·3-s + 0.696·4-s + 1.27·6-s + 0.582·7-s + 0.395·8-s − 0.0402·9-s + 0.764·11-s − 0.682·12-s + 1.76·13-s − 0.758·14-s − 1.21·16-s − 0.859·17-s + 0.0524·18-s + 0.163·19-s − 0.570·21-s − 0.995·22-s + 0.347·23-s − 0.387·24-s − 2.30·26-s + 1.01·27-s + 0.405·28-s − 1.53·29-s + 1.89·31-s + 1.18·32-s − 0.749·33-s + 1.11·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 + 1.84T + 2T^{2} \) |
| 3 | \( 1 + 1.69T + 3T^{2} \) |
| 7 | \( 1 - 1.54T + 7T^{2} \) |
| 11 | \( 1 - 2.53T + 11T^{2} \) |
| 13 | \( 1 - 6.36T + 13T^{2} \) |
| 17 | \( 1 + 3.54T + 17T^{2} \) |
| 19 | \( 1 - 0.713T + 19T^{2} \) |
| 23 | \( 1 - 1.66T + 23T^{2} \) |
| 29 | \( 1 + 8.23T + 29T^{2} \) |
| 31 | \( 1 - 10.5T + 31T^{2} \) |
| 37 | \( 1 + 0.496T + 37T^{2} \) |
| 41 | \( 1 + 8.92T + 41T^{2} \) |
| 43 | \( 1 - 1.50T + 43T^{2} \) |
| 47 | \( 1 - 6.94T + 47T^{2} \) |
| 53 | \( 1 + 9.87T + 53T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 + 9.13T + 61T^{2} \) |
| 67 | \( 1 + 1.85T + 67T^{2} \) |
| 71 | \( 1 + 9.47T + 71T^{2} \) |
| 73 | \( 1 + 8.88T + 73T^{2} \) |
| 79 | \( 1 + 6.12T + 79T^{2} \) |
| 83 | \( 1 + 0.835T + 83T^{2} \) |
| 89 | \( 1 - 2.27T + 89T^{2} \) |
| 97 | \( 1 + 7.56T + 97T^{2} \) |
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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
−7.908995054557943175255057510357, −7.05342085290604124084497256502, −6.34311510421423605459782481626, −5.88914637037995298383248560758, −4.80625571715996467640406049261, −4.25404461943419426994134581310, −3.11577740386111299472060235934, −1.68671407914604073859012630061, −1.15468115527073731507429438315, 0,
1.15468115527073731507429438315, 1.68671407914604073859012630061, 3.11577740386111299472060235934, 4.25404461943419426994134581310, 4.80625571715996467640406049261, 5.88914637037995298383248560758, 6.34311510421423605459782481626, 7.05342085290604124084497256502, 7.908995054557943175255057510357
