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
−10.39752056417284564200039685411, −9.413153470421490451716819562831, −8.446711060340473634434965915963, −7.74168347229201480685589194695, −6.89889128135846713628218821821, −6.41912833207372144341487067994, −5.01056250342754943947983008074, −4.06418091952156900545821215912, −2.67288254076284268713814859926, −1.72128082924341836682443571602,
0.24399902680607335928413778972, 2.44057478019176540824317934035, 3.75194825977251030991673795239, 4.22105392031515257806602072715, 5.50644505198221759450641373105, 6.08972356542199230241496514557, 7.55101064028644912551552740674, 8.426418549138449507070944013397, 8.893632250467615455628010731437, 9.865823949964492969621253480271