| L(s) = 1 | + (−1.40 − 0.179i)2-s + (1.93 + 0.502i)4-s + (0.847 − 2.32i)5-s + (−4.59 − 0.810i)7-s + (−2.62 − 1.05i)8-s + (−1.60 + 3.11i)10-s + (−2.23 + 0.812i)11-s + (−1.53 + 1.28i)13-s + (6.30 + 1.95i)14-s + (3.49 + 1.94i)16-s + (−1.59 + 0.920i)17-s + (−1.56 − 0.902i)19-s + (2.81 − 4.08i)20-s + (3.27 − 0.740i)22-s + (−0.496 − 2.81i)23-s + ⋯ |
| L(s) = 1 | + (−0.991 − 0.126i)2-s + (0.967 + 0.251i)4-s + (0.379 − 1.04i)5-s + (−1.73 − 0.306i)7-s + (−0.928 − 0.371i)8-s + (−0.508 + 0.985i)10-s + (−0.673 + 0.244i)11-s + (−0.425 + 0.357i)13-s + (1.68 + 0.523i)14-s + (0.873 + 0.486i)16-s + (−0.386 + 0.223i)17-s + (−0.358 − 0.207i)19-s + (0.628 − 0.913i)20-s + (0.698 − 0.157i)22-s + (−0.103 − 0.586i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.104i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.104i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0126693 - 0.242329i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0126693 - 0.242329i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.40 + 0.179i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-0.847 + 2.32i)T + (-3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (4.59 + 0.810i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (2.23 - 0.812i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (1.53 - 1.28i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (1.59 - 0.920i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.56 + 0.902i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.496 + 2.81i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (4.19 - 5.00i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (0.704 - 0.124i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (5.60 + 9.71i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (4.52 + 5.39i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.206 - 0.568i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.587 + 3.32i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 5.16iT - 53T^{2} \) |
| 59 | \( 1 + (-7.69 - 2.79i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.01 + 5.75i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (0.427 + 0.509i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (0.933 + 1.61i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.519 + 0.899i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-8.75 + 10.4i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (2.22 + 1.86i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-9.13 - 5.27i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (14.6 - 5.31i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.80943722558853803910937992481, −10.08037091525517215781400459243, −9.243008477953743096514730734856, −8.714155734954071038152605914924, −7.31712919213047198731383860754, −6.54696401121409493028991084280, −5.34134189301734358250515757586, −3.65044549909307072065470195035, −2.17964439913255024554405872587, −0.21280782381346357256287130104,
2.48407132207769541256070923197, 3.25718783080218277313488987463, 5.66279716814195272528900487504, 6.47582002492134423625030920156, 7.16903932756052760945122056756, 8.350259606876167696061846693550, 9.570377527741790988578980914097, 10.01022680983192656979834970511, 10.78801078933950578603730708845, 11.85194371849109205657783335512
