| L(s) = 1 | − 1.73·2-s + 0.999·4-s + 5-s + 2·7-s + 1.73·8-s − 1.73·10-s + 11-s − 1.46·13-s − 3.46·14-s − 5·16-s − 1.46·19-s + 0.999·20-s − 1.73·22-s + 6.92·23-s + 25-s + 2.53·26-s + 1.99·28-s − 3.46·29-s + 2.92·31-s + 5.19·32-s + 2·35-s + 8.92·37-s + 2.53·38-s + 1.73·40-s + 3.46·41-s + 8.92·43-s + 0.999·44-s + ⋯ |
| L(s) = 1 | − 1.22·2-s + 0.499·4-s + 0.447·5-s + 0.755·7-s + 0.612·8-s − 0.547·10-s + 0.301·11-s − 0.406·13-s − 0.925·14-s − 1.25·16-s − 0.335·19-s + 0.223·20-s − 0.369·22-s + 1.44·23-s + 0.200·25-s + 0.497·26-s + 0.377·28-s − 0.643·29-s + 0.525·31-s + 0.918·32-s + 0.338·35-s + 1.46·37-s + 0.411·38-s + 0.273·40-s + 0.541·41-s + 1.36·43-s + 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8727559747\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8727559747\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + 1.73T + 2T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 13 | \( 1 + 1.46T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 1.46T + 19T^{2} \) |
| 23 | \( 1 - 6.92T + 23T^{2} \) |
| 29 | \( 1 + 3.46T + 29T^{2} \) |
| 31 | \( 1 - 2.92T + 31T^{2} \) |
| 37 | \( 1 - 8.92T + 37T^{2} \) |
| 41 | \( 1 - 3.46T + 41T^{2} \) |
| 43 | \( 1 - 8.92T + 43T^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 - 12.9T + 53T^{2} \) |
| 59 | \( 1 + 6.92T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 + 15.4T + 83T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−10.89127882137597332044623208978, −9.817864852424828595250635592134, −9.235840824968697107592518496619, −8.371418722385280306901806747407, −7.57348372324036685001824067013, −6.64737652328587879582815975106, −5.29469398389400579696305945589, −4.29056875685508697222209046026, −2.42160994809114831712228434286, −1.10925676578488351934899627340,
1.10925676578488351934899627340, 2.42160994809114831712228434286, 4.29056875685508697222209046026, 5.29469398389400579696305945589, 6.64737652328587879582815975106, 7.57348372324036685001824067013, 8.371418722385280306901806747407, 9.235840824968697107592518496619, 9.817864852424828595250635592134, 10.89127882137597332044623208978
