| L(s) = 1 | − 0.257·2-s + 1.45·3-s − 1.93·4-s − 1.48·5-s − 0.374·6-s − 1.22·7-s + 1.01·8-s − 0.887·9-s + 0.383·10-s − 0.271·11-s − 2.81·12-s + 1.75·13-s + 0.314·14-s − 2.16·15-s + 3.60·16-s − 2.25·17-s + 0.228·18-s + 7.74·19-s + 2.87·20-s − 1.77·21-s + 0.0699·22-s − 6.76·23-s + 1.47·24-s − 2.78·25-s − 0.451·26-s − 5.65·27-s + 2.36·28-s + ⋯ |
| L(s) = 1 | − 0.182·2-s + 0.839·3-s − 0.966·4-s − 0.665·5-s − 0.152·6-s − 0.462·7-s + 0.358·8-s − 0.295·9-s + 0.121·10-s − 0.0819·11-s − 0.811·12-s + 0.486·13-s + 0.0841·14-s − 0.558·15-s + 0.901·16-s − 0.545·17-s + 0.0538·18-s + 1.77·19-s + 0.643·20-s − 0.387·21-s + 0.0149·22-s − 1.40·23-s + 0.300·24-s − 0.557·25-s − 0.0885·26-s − 1.08·27-s + 0.446·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.003134908\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.003134908\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 8011 | \( 1+O(T) \) |
| good | 2 | \( 1 + 0.257T + 2T^{2} \) |
| 3 | \( 1 - 1.45T + 3T^{2} \) |
| 5 | \( 1 + 1.48T + 5T^{2} \) |
| 7 | \( 1 + 1.22T + 7T^{2} \) |
| 11 | \( 1 + 0.271T + 11T^{2} \) |
| 13 | \( 1 - 1.75T + 13T^{2} \) |
| 17 | \( 1 + 2.25T + 17T^{2} \) |
| 19 | \( 1 - 7.74T + 19T^{2} \) |
| 23 | \( 1 + 6.76T + 23T^{2} \) |
| 29 | \( 1 - 4.24T + 29T^{2} \) |
| 31 | \( 1 + 1.72T + 31T^{2} \) |
| 37 | \( 1 + 8.48T + 37T^{2} \) |
| 41 | \( 1 + 2.86T + 41T^{2} \) |
| 43 | \( 1 - 1.13T + 43T^{2} \) |
| 47 | \( 1 - 4.39T + 47T^{2} \) |
| 53 | \( 1 + 3.47T + 53T^{2} \) |
| 59 | \( 1 + 6.20T + 59T^{2} \) |
| 61 | \( 1 - 12.7T + 61T^{2} \) |
| 67 | \( 1 - 10.8T + 67T^{2} \) |
| 71 | \( 1 - 5.72T + 71T^{2} \) |
| 73 | \( 1 - 13.7T + 73T^{2} \) |
| 79 | \( 1 + 15.4T + 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 - 13.9T + 89T^{2} \) |
| 97 | \( 1 + 11.2T + 97T^{2} \) |
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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
−7.974216546291810725003724324197, −7.47729618592696316159534385850, −6.52852973653426031110164671060, −5.62590178634350344003306374534, −5.02578800859957399599911839092, −3.93108562734919931406200812358, −3.65470358672397237757335916138, −2.89309065525119877567949456363, −1.75948556035342595742704247964, −0.48612665206569034723712748759,
0.48612665206569034723712748759, 1.75948556035342595742704247964, 2.89309065525119877567949456363, 3.65470358672397237757335916138, 3.93108562734919931406200812358, 5.02578800859957399599911839092, 5.62590178634350344003306374534, 6.52852973653426031110164671060, 7.47729618592696316159534385850, 7.974216546291810725003724324197
