L(s) = 1 | + (2.59 − 1.50i)3-s + (−2.59 − 2.59i)5-s − 7.30i·7-s + (4.47 − 7.81i)9-s + (11.3 + 11.3i)11-s + (0.746 + 0.746i)13-s + (−10.6 − 2.83i)15-s − 6.67i·17-s + (−22.1 − 22.1i)19-s + (−10.9 − 18.9i)21-s − 21.4·23-s − 11.4i·25-s + (−0.153 − 26.9i)27-s + (1.54 − 1.54i)29-s − 14.6·31-s + ⋯ |
L(s) = 1 | + (0.865 − 0.501i)3-s + (−0.519 − 0.519i)5-s − 1.04i·7-s + (0.496 − 0.867i)9-s + (1.02 + 1.02i)11-s + (0.0574 + 0.0574i)13-s + (−0.710 − 0.188i)15-s − 0.392i·17-s + (−1.16 − 1.16i)19-s + (−0.523 − 0.902i)21-s − 0.932·23-s − 0.459i·25-s + (−0.00567 − 0.999i)27-s + (0.0531 − 0.0531i)29-s − 0.471·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.249 + 0.968i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.249 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.19597 - 1.54279i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.19597 - 1.54279i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.59 + 1.50i)T \) |
good | 5 | \( 1 + (2.59 + 2.59i)T + 25iT^{2} \) |
| 7 | \( 1 + 7.30iT - 49T^{2} \) |
| 11 | \( 1 + (-11.3 - 11.3i)T + 121iT^{2} \) |
| 13 | \( 1 + (-0.746 - 0.746i)T + 169iT^{2} \) |
| 17 | \( 1 + 6.67iT - 289T^{2} \) |
| 19 | \( 1 + (22.1 + 22.1i)T + 361iT^{2} \) |
| 23 | \( 1 + 21.4T + 529T^{2} \) |
| 29 | \( 1 + (-1.54 + 1.54i)T - 841iT^{2} \) |
| 31 | \( 1 + 14.6T + 961T^{2} \) |
| 37 | \( 1 + (-50.1 + 50.1i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 - 15.0T + 1.68e3T^{2} \) |
| 43 | \( 1 + (26.3 - 26.3i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 - 36.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-50.9 - 50.9i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (12.1 + 12.1i)T + 3.48e3iT^{2} \) |
| 61 | \( 1 + (-27.5 - 27.5i)T + 3.72e3iT^{2} \) |
| 67 | \( 1 + (-4.84 - 4.84i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 - 74.9T + 5.04e3T^{2} \) |
| 73 | \( 1 - 3.47iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 103.T + 6.24e3T^{2} \) |
| 83 | \( 1 + (31.7 - 31.7i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 - 78.2T + 7.92e3T^{2} \) |
| 97 | \( 1 + 61.5T + 9.40e3T^{2} \) |
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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
−10.87655512769640493580452666170, −9.692500141697563438776759697046, −9.014231919680626299475384825067, −7.980425978869912446246982385631, −7.21776397726315551092397209148, −6.42991889451752569082975238791, −4.40776100463164273652950047397, −3.99894793062633627187472467555, −2.27173218779678413416540058141, −0.802539649357753360204362816719,
2.01049445344633146170850902611, 3.35207013651714682988300587196, 4.06769285920214802809779897634, 5.64163620660273569741039917792, 6.61814739248475699583447412784, 8.068191038628073027632768613144, 8.511420297979630433019559747507, 9.449548449046982934792845323395, 10.43276803591504756472420101247, 11.35718322080001391400198221058