| L(s) = 1 | − 2.69·2-s + 1.61·3-s + 5.28·4-s + 2.80·5-s − 4.36·6-s + 2.73·7-s − 8.87·8-s − 0.389·9-s − 7.57·10-s + 3.22·11-s + 8.54·12-s + 4.46·13-s − 7.38·14-s + 4.53·15-s + 13.3·16-s + 5.42·17-s + 1.05·18-s − 2.28·19-s + 14.8·20-s + 4.42·21-s − 8.70·22-s − 6.19·23-s − 14.3·24-s + 2.87·25-s − 12.0·26-s − 5.47·27-s + 14.4·28-s + ⋯ |
| L(s) = 1 | − 1.90·2-s + 0.932·3-s + 2.64·4-s + 1.25·5-s − 1.78·6-s + 1.03·7-s − 3.13·8-s − 0.129·9-s − 2.39·10-s + 0.972·11-s + 2.46·12-s + 1.23·13-s − 1.97·14-s + 1.17·15-s + 3.34·16-s + 1.31·17-s + 0.248·18-s − 0.524·19-s + 3.31·20-s + 0.965·21-s − 1.85·22-s − 1.29·23-s − 2.92·24-s + 0.575·25-s − 2.36·26-s − 1.05·27-s + 2.73·28-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
| \(L(1)\) |
\(\approx\) |
\(2.120679351\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.120679351\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.69T + 2T^{2} \) |
| 3 | \( 1 - 1.61T + 3T^{2} \) |
| 5 | \( 1 - 2.80T + 5T^{2} \) |
| 7 | \( 1 - 2.73T + 7T^{2} \) |
| 11 | \( 1 - 3.22T + 11T^{2} \) |
| 13 | \( 1 - 4.46T + 13T^{2} \) |
| 17 | \( 1 - 5.42T + 17T^{2} \) |
| 19 | \( 1 + 2.28T + 19T^{2} \) |
| 23 | \( 1 + 6.19T + 23T^{2} \) |
| 29 | \( 1 - 0.0185T + 29T^{2} \) |
| 31 | \( 1 - 5.36T + 31T^{2} \) |
| 37 | \( 1 + 7.87T + 37T^{2} \) |
| 41 | \( 1 - 2.29T + 41T^{2} \) |
| 43 | \( 1 + 0.973T + 43T^{2} \) |
| 47 | \( 1 - 9.69T + 47T^{2} \) |
| 53 | \( 1 + 7.73T + 53T^{2} \) |
| 59 | \( 1 - 12.1T + 59T^{2} \) |
| 61 | \( 1 + 10.2T + 61T^{2} \) |
| 67 | \( 1 - 6.76T + 67T^{2} \) |
| 71 | \( 1 - 15.3T + 71T^{2} \) |
| 73 | \( 1 + 5.09T + 73T^{2} \) |
| 79 | \( 1 + 5.64T + 79T^{2} \) |
| 83 | \( 1 + 5.87T + 83T^{2} \) |
| 89 | \( 1 - 0.696T + 89T^{2} \) |
| 97 | \( 1 - 8.33T + 97T^{2} \) |
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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
−8.147487519042530850001648086835, −7.56756419038165674412369808663, −6.57329173705848763255323458128, −6.06591540373736501116105542145, −5.46284627061467062370559631454, −3.94590972398838766042771870057, −3.10616257082431958650115060594, −2.11118703013024920801433741579, −1.72640919192590163368505803001, −0.989531643185154670349609037686,
0.989531643185154670349609037686, 1.72640919192590163368505803001, 2.11118703013024920801433741579, 3.10616257082431958650115060594, 3.94590972398838766042771870057, 5.46284627061467062370559631454, 6.06591540373736501116105542145, 6.57329173705848763255323458128, 7.56756419038165674412369808663, 8.147487519042530850001648086835
