L(s) = 1 | + (0.158 + 1.72i)3-s + (−1.22 − 0.707i)5-s + 0.267·7-s + (−2.94 + 0.548i)9-s − 5.27i·11-s + (−0.232 + 0.133i)13-s + (1.02 − 2.22i)15-s + (−4.24 − 2.44i)17-s + (−1.73 + 4i)19-s + (0.0425 + 0.462i)21-s + (−4.57 + 2.63i)23-s + (−1.50 − 2.59i)25-s + (−1.41 − 4.99i)27-s + (−1.03 − 1.79i)29-s − 2.46i·31-s + ⋯ |
L(s) = 1 | + (0.0917 + 0.995i)3-s + (−0.547 − 0.316i)5-s + 0.101·7-s + (−0.983 + 0.182i)9-s − 1.59i·11-s + (−0.0643 + 0.0371i)13-s + (0.264 − 0.574i)15-s + (−1.02 − 0.594i)17-s + (−0.397 + 0.917i)19-s + (0.00929 + 0.100i)21-s + (−0.953 + 0.550i)23-s + (−0.300 − 0.519i)25-s + (−0.272 − 0.962i)27-s + (−0.192 − 0.332i)29-s − 0.442i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.217 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.217 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.343468 - 0.428259i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.343468 - 0.428259i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.158 - 1.72i)T \) |
| 19 | \( 1 + (1.73 - 4i)T \) |
good | 5 | \( 1 + (1.22 + 0.707i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 - 0.267T + 7T^{2} \) |
| 11 | \( 1 + 5.27iT - 11T^{2} \) |
| 13 | \( 1 + (0.232 - 0.133i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (4.24 + 2.44i)T + (8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (4.57 - 2.63i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.03 + 1.79i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 2.46iT - 31T^{2} \) |
| 37 | \( 1 + 7.73iT - 37T^{2} \) |
| 41 | \( 1 + (-2.82 + 4.89i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.86 + 4.96i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.656 - 0.378i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.46 - 9.46i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.60 + 9.71i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.23 - 9.06i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.866 + 0.5i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (6.69 - 11.5i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.5 + 2.59i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (9.06 + 5.23i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 2.07iT - 83T^{2} \) |
| 89 | \( 1 + (3.67 + 6.36i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.464 + 0.267i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.865823949964492969621253480271, −8.893632250467615455628010731437, −8.426418549138449507070944013397, −7.55101064028644912551552740674, −6.08972356542199230241496514557, −5.50644505198221759450641373105, −4.22105392031515257806602072715, −3.75194825977251030991673795239, −2.44057478019176540824317934035, −0.24399902680607335928413778972,
1.72128082924341836682443571602, 2.67288254076284268713814859926, 4.06418091952156900545821215912, 5.01056250342754943947983008074, 6.41912833207372144341487067994, 6.89889128135846713628218821821, 7.74168347229201480685589194695, 8.446711060340473634434965915963, 9.413153470421490451716819562831, 10.39752056417284564200039685411