| L(s) = 1 | − 0.512·2-s − 1.73·4-s + 1.76·5-s − 7-s + 1.91·8-s − 0.906·10-s − 4.52·11-s + 4.50·13-s + 0.512·14-s + 2.49·16-s + 5.81·17-s + 6.77·19-s − 3.07·20-s + 2.31·22-s − 0.708·23-s − 1.86·25-s − 2.30·26-s + 1.73·28-s + 8.00·29-s + 8.44·31-s − 5.10·32-s − 2.97·34-s − 1.76·35-s + 1.40·37-s − 3.47·38-s + 3.38·40-s − 3.06·41-s + ⋯ |
| L(s) = 1 | − 0.362·2-s − 0.868·4-s + 0.791·5-s − 0.377·7-s + 0.676·8-s − 0.286·10-s − 1.36·11-s + 1.24·13-s + 0.136·14-s + 0.623·16-s + 1.41·17-s + 1.55·19-s − 0.687·20-s + 0.493·22-s − 0.147·23-s − 0.373·25-s − 0.452·26-s + 0.328·28-s + 1.48·29-s + 1.51·31-s − 0.902·32-s − 0.511·34-s − 0.299·35-s + 0.231·37-s − 0.563·38-s + 0.535·40-s − 0.478·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.676219071\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.676219071\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 + 0.512T + 2T^{2} \) |
| 5 | \( 1 - 1.76T + 5T^{2} \) |
| 11 | \( 1 + 4.52T + 11T^{2} \) |
| 13 | \( 1 - 4.50T + 13T^{2} \) |
| 17 | \( 1 - 5.81T + 17T^{2} \) |
| 19 | \( 1 - 6.77T + 19T^{2} \) |
| 23 | \( 1 + 0.708T + 23T^{2} \) |
| 29 | \( 1 - 8.00T + 29T^{2} \) |
| 31 | \( 1 - 8.44T + 31T^{2} \) |
| 37 | \( 1 - 1.40T + 37T^{2} \) |
| 41 | \( 1 + 3.06T + 41T^{2} \) |
| 43 | \( 1 + 3.92T + 43T^{2} \) |
| 47 | \( 1 - 9.77T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 - 10.1T + 59T^{2} \) |
| 61 | \( 1 + 9.39T + 61T^{2} \) |
| 67 | \( 1 + 9.67T + 67T^{2} \) |
| 71 | \( 1 - 7.71T + 71T^{2} \) |
| 73 | \( 1 + 14.8T + 73T^{2} \) |
| 79 | \( 1 - 13.3T + 79T^{2} \) |
| 83 | \( 1 + 15.4T + 83T^{2} \) |
| 89 | \( 1 + 4.77T + 89T^{2} \) |
| 97 | \( 1 - 10.7T + 97T^{2} \) |
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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
−8.007964245674857411652258066107, −7.36323007162765493775170180161, −6.27880108810493744856751297485, −5.68362611559112802077704143951, −5.19336332338442930933542213256, −4.38206701964609466456371246719, −3.32589539774817441800380875589, −2.84381625798372055976785268267, −1.46869815383634874386207450886, −0.75313425124844941374057735141,
0.75313425124844941374057735141, 1.46869815383634874386207450886, 2.84381625798372055976785268267, 3.32589539774817441800380875589, 4.38206701964609466456371246719, 5.19336332338442930933542213256, 5.68362611559112802077704143951, 6.27880108810493744856751297485, 7.36323007162765493775170180161, 8.007964245674857411652258066107
