| L(s) = 1 | + 1.73·3-s + 2.82i·5-s − 4.89i·7-s + 2.99·9-s + 3.46·11-s + 4.89i·15-s − 8.48i·21-s − 3.00·25-s + 5.19·27-s − 2.82i·29-s + 4.89i·31-s + 5.99·33-s + 13.8·35-s + 8.48i·45-s − 16.9·49-s + ⋯ |
| L(s) = 1 | + 1.00·3-s + 1.26i·5-s − 1.85i·7-s + 0.999·9-s + 1.04·11-s + 1.26i·15-s − 1.85i·21-s − 0.600·25-s + 1.00·27-s − 0.525i·29-s + 0.879i·31-s + 1.04·33-s + 2.34·35-s + 1.26i·45-s − 2.42·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.31831\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.31831\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73T \) |
| good | 5 | \( 1 - 2.82iT - 5T^{2} \) |
| 7 | \( 1 + 4.89iT - 7T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 2.82iT - 29T^{2} \) |
| 31 | \( 1 - 4.89iT - 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 14.1iT - 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 14T + 73T^{2} \) |
| 79 | \( 1 + 14.6iT - 79T^{2} \) |
| 83 | \( 1 + 17.3T + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−10.32909760570079221481890994929, −9.629837680480124720927457136546, −8.578577146622862968058196661293, −7.45213944983189682014563506023, −7.11533425774731656430088601478, −6.33069846030775356871813541348, −4.41892135810656244908119157414, −3.75655952466733903797859199920, −2.89904344157900136338280407218, −1.36425066558286993075187059679,
1.49911110788580926479746368750, 2.54947405039769471395279578509, 3.83292364910677482169371624067, 4.89664487839473580130220161652, 5.76290138329858101247113134895, 6.88286165107540979374100367524, 8.257482664673325897802758815756, 8.616706418367673153960466053785, 9.279583949511853532075757120473, 9.812357515712544434123705176143
