| L(s) = 1 | − 1.03·2-s − 2.03·3-s − 0.931·4-s − 0.515·5-s + 2.09·6-s − 3.45·7-s + 3.03·8-s + 1.12·9-s + 0.533·10-s + 11-s + 1.89·12-s − 1.48·13-s + 3.57·14-s + 1.04·15-s − 1.26·16-s − 5.99·17-s − 1.15·18-s − 2.55·19-s + 0.480·20-s + 7.02·21-s − 1.03·22-s − 7.11·23-s − 6.15·24-s − 4.73·25-s + 1.53·26-s + 3.81·27-s + 3.22·28-s + ⋯ |
| L(s) = 1 | − 0.730·2-s − 1.17·3-s − 0.465·4-s − 0.230·5-s + 0.856·6-s − 1.30·7-s + 1.07·8-s + 0.374·9-s + 0.168·10-s + 0.301·11-s + 0.546·12-s − 0.412·13-s + 0.955·14-s + 0.270·15-s − 0.316·16-s − 1.45·17-s − 0.273·18-s − 0.585·19-s + 0.107·20-s + 1.53·21-s − 0.220·22-s − 1.48·23-s − 1.25·24-s − 0.946·25-s + 0.301·26-s + 0.733·27-s + 0.609·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.05305685481\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05305685481\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 131 | \( 1 + T \) |
| good | 2 | \( 1 + 1.03T + 2T^{2} \) |
| 3 | \( 1 + 2.03T + 3T^{2} \) |
| 5 | \( 1 + 0.515T + 5T^{2} \) |
| 7 | \( 1 + 3.45T + 7T^{2} \) |
| 13 | \( 1 + 1.48T + 13T^{2} \) |
| 17 | \( 1 + 5.99T + 17T^{2} \) |
| 19 | \( 1 + 2.55T + 19T^{2} \) |
| 23 | \( 1 + 7.11T + 23T^{2} \) |
| 29 | \( 1 + 5.28T + 29T^{2} \) |
| 31 | \( 1 + 7.74T + 31T^{2} \) |
| 37 | \( 1 - 1.60T + 37T^{2} \) |
| 41 | \( 1 + 2.10T + 41T^{2} \) |
| 43 | \( 1 - 7.25T + 43T^{2} \) |
| 47 | \( 1 + 8.70T + 47T^{2} \) |
| 53 | \( 1 + 8.54T + 53T^{2} \) |
| 59 | \( 1 - 8.26T + 59T^{2} \) |
| 61 | \( 1 - 4.86T + 61T^{2} \) |
| 67 | \( 1 - 1.50T + 67T^{2} \) |
| 71 | \( 1 - 7.28T + 71T^{2} \) |
| 73 | \( 1 + 15.6T + 73T^{2} \) |
| 79 | \( 1 - 1.90T + 79T^{2} \) |
| 83 | \( 1 - 13.2T + 83T^{2} \) |
| 89 | \( 1 + 6.04T + 89T^{2} \) |
| 97 | \( 1 + 17.2T + 97T^{2} \) |
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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
−9.577545083487962261501326979188, −8.934208054924645165629011802136, −7.976733308027789571028935869423, −7.01582336893646756908850592717, −6.30175681504077995371353327528, −5.55125477853338557609951736595, −4.44536889431878955580286573825, −3.71570019490984182247313449180, −2.02816446288192344079150595491, −0.18536154876094824485617443665,
0.18536154876094824485617443665, 2.02816446288192344079150595491, 3.71570019490984182247313449180, 4.44536889431878955580286573825, 5.55125477853338557609951736595, 6.30175681504077995371353327528, 7.01582336893646756908850592717, 7.976733308027789571028935869423, 8.934208054924645165629011802136, 9.577545083487962261501326979188
