| L(s) = 1 | + (−10.1 − 11.8i)3-s − 19.3i·5-s − 82.7i·7-s + (−38.1 + 239. i)9-s − 332.·11-s + 1.17e3·13-s + (−229. + 195. i)15-s − 1.98e3i·17-s + 1.34e3i·19-s + (−981. + 837. i)21-s + 1.83e3·23-s + 2.75e3·25-s + (3.23e3 − 1.97e3i)27-s − 1.92e3i·29-s − 5.80e3i·31-s + ⋯ |
| L(s) = 1 | + (−0.649 − 0.760i)3-s − 0.345i·5-s − 0.638i·7-s + (−0.157 + 0.987i)9-s − 0.828·11-s + 1.92·13-s + (−0.262 + 0.224i)15-s − 1.66i·17-s + 0.855i·19-s + (−0.485 + 0.414i)21-s + 0.722·23-s + 0.880·25-s + (0.853 − 0.521i)27-s − 0.424i·29-s − 1.08i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.760 + 0.649i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.760 + 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.471733617\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.471733617\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (10.1 + 11.8i)T \) |
| good | 5 | \( 1 + 19.3iT - 3.12e3T^{2} \) |
| 7 | \( 1 + 82.7iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 332.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 1.17e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.98e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 1.34e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 1.83e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.92e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 5.80e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 6.80e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 4.15e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 8.37e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 1.80e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.27e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 1.86e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.95e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.09e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 6.98e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.70e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.05e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + 9.63e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.79e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 - 4.83e4T + 8.58e9T^{2} \) |
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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
−10.46935550365953771269987477340, −9.175622726062273159911991597646, −8.089037901726073174173781509351, −7.37188502740605931864406963419, −6.30712083123002581385778998755, −5.44116458351516938273468740667, −4.34580980500408334606828603564, −2.85025618933672104178267252103, −1.29437251521083522066458895962, −0.48840008988058303932972320686,
1.16586787196170854067991484310, 2.91801727898070765740009796327, 3.92375303084419204487902522597, 5.16802526036903563701240696783, 5.99441341846034962409086432495, 6.81789725500417281461750853577, 8.468364140500759623531883561837, 8.898718728479235375451537671009, 10.28604098922116885489296981054, 10.82480674615516534556861990178
