| L(s) = 1 | + (1.37 − 1.44i)2-s + (−0.193 − 3.99i)4-s − 5.95i·5-s + 4.72i·7-s + (−6.05 − 5.23i)8-s + (−8.62 − 8.21i)10-s − 13.8·11-s + 4.65i·13-s + (6.83 + 6.51i)14-s + (−15.9 + 1.54i)16-s − 21.5·17-s + 3.83·19-s + (−23.7 + 1.15i)20-s + (−19.1 + 20.1i)22-s − 34.6i·23-s + ⋯ |
| L(s) = 1 | + (0.689 − 0.724i)2-s + (−0.0484 − 0.998i)4-s − 1.19i·5-s + 0.674i·7-s + (−0.756 − 0.653i)8-s + (−0.862 − 0.821i)10-s − 1.26·11-s + 0.358i·13-s + (0.488 + 0.465i)14-s + (−0.995 + 0.0966i)16-s − 1.26·17-s + 0.201·19-s + (−1.18 + 0.0576i)20-s + (−0.871 + 0.914i)22-s − 1.50i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.756 - 0.653i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.756 - 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.030082447\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.030082447\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.37 + 1.44i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 5.95iT - 25T^{2} \) |
| 7 | \( 1 - 4.72iT - 49T^{2} \) |
| 11 | \( 1 + 13.8T + 121T^{2} \) |
| 13 | \( 1 - 4.65iT - 169T^{2} \) |
| 17 | \( 1 + 21.5T + 289T^{2} \) |
| 19 | \( 1 - 3.83T + 361T^{2} \) |
| 23 | \( 1 + 34.6iT - 529T^{2} \) |
| 29 | \( 1 - 45.4iT - 841T^{2} \) |
| 31 | \( 1 + 36.8iT - 961T^{2} \) |
| 37 | \( 1 - 36.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 20.4T + 1.68e3T^{2} \) |
| 43 | \( 1 + 7.01T + 1.84e3T^{2} \) |
| 47 | \( 1 + 61.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 58.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 22.0T + 3.48e3T^{2} \) |
| 61 | \( 1 - 54.3iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 113.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 8.30iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 114.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 54.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 31.2T + 6.88e3T^{2} \) |
| 89 | \( 1 + 47.7T + 7.92e3T^{2} \) |
| 97 | \( 1 - 57.5T + 9.40e3T^{2} \) |
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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
−9.988134472191662183248045649793, −8.842371102146866601278482786157, −8.552235358228709284045743918793, −6.95552036103648439771546363099, −5.81906820961327906004623815878, −4.98222247899793612869074813473, −4.40516350411355285481826162552, −2.87181075514755461904536756191, −1.86605762524461291564079811399, −0.27298827118785760326654385948,
2.47813224559795858905033266484, 3.36373099261058963996760008269, 4.45691104360995671215002387614, 5.55582958843844192515417800857, 6.45722596213115391176679712926, 7.39129106242844471359680368996, 7.74834084295791675699860765611, 9.001699264429971388353943785197, 10.23158122018957793460751424619, 10.90396140908852305174329715494
