| L(s) = 1 | + (−1.27 − 3.92i)2-s + (0.664 − 5.15i)3-s + (−7.33 + 5.32i)4-s + (0.616 + 0.200i)5-s + (−21.0 + 3.96i)6-s + (8.50 + 11.7i)7-s + (3.54 + 2.57i)8-s + (−26.1 − 6.84i)9-s − 2.67i·10-s + (16.7 − 32.3i)11-s + (22.5 + 41.3i)12-s + (54.2 − 17.6i)13-s + (35.1 − 48.3i)14-s + (1.44 − 3.04i)15-s + (−16.8 + 51.7i)16-s + (−5.39 + 16.5i)17-s + ⋯ |
| L(s) = 1 | + (−0.451 − 1.38i)2-s + (0.127 − 0.991i)3-s + (−0.916 + 0.665i)4-s + (0.0551 + 0.0179i)5-s + (−1.43 + 0.270i)6-s + (0.459 + 0.632i)7-s + (0.156 + 0.113i)8-s + (−0.967 − 0.253i)9-s − 0.0846i·10-s + (0.459 − 0.887i)11-s + (0.543 + 0.993i)12-s + (1.15 − 0.376i)13-s + (0.670 − 0.923i)14-s + (0.0248 − 0.0523i)15-s + (−0.262 + 0.808i)16-s + (−0.0769 + 0.236i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.936 + 0.350i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.936 + 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.182575 - 1.00785i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.182575 - 1.00785i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.664 + 5.15i)T \) |
| 11 | \( 1 + (-16.7 + 32.3i)T \) |
| good | 2 | \( 1 + (1.27 + 3.92i)T + (-6.47 + 4.70i)T^{2} \) |
| 5 | \( 1 + (-0.616 - 0.200i)T + (101. + 73.4i)T^{2} \) |
| 7 | \( 1 + (-8.50 - 11.7i)T + (-105. + 326. i)T^{2} \) |
| 13 | \( 1 + (-54.2 + 17.6i)T + (1.77e3 - 1.29e3i)T^{2} \) |
| 17 | \( 1 + (5.39 - 16.5i)T + (-3.97e3 - 2.88e3i)T^{2} \) |
| 19 | \( 1 + (-62.1 + 85.4i)T + (-2.11e3 - 6.52e3i)T^{2} \) |
| 23 | \( 1 - 145. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (-26.1 + 19.0i)T + (7.53e3 - 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-80.1 - 246. i)T + (-2.41e4 + 1.75e4i)T^{2} \) |
| 37 | \( 1 + (-120. + 87.7i)T + (1.56e4 - 4.81e4i)T^{2} \) |
| 41 | \( 1 + (-246. - 179. i)T + (2.12e4 + 6.55e4i)T^{2} \) |
| 43 | \( 1 - 267. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (147. - 203. i)T + (-3.20e4 - 9.87e4i)T^{2} \) |
| 53 | \( 1 + (225. - 73.3i)T + (1.20e5 - 8.75e4i)T^{2} \) |
| 59 | \( 1 + (394. + 543. i)T + (-6.34e4 + 1.95e5i)T^{2} \) |
| 61 | \( 1 + (-188. - 61.1i)T + (1.83e5 + 1.33e5i)T^{2} \) |
| 67 | \( 1 + 557.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-226. - 73.6i)T + (2.89e5 + 2.10e5i)T^{2} \) |
| 73 | \( 1 + (420. + 578. i)T + (-1.20e5 + 3.69e5i)T^{2} \) |
| 79 | \( 1 + (973. - 316. i)T + (3.98e5 - 2.89e5i)T^{2} \) |
| 83 | \( 1 + (278. - 857. i)T + (-4.62e5 - 3.36e5i)T^{2} \) |
| 89 | \( 1 - 165. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + (-158. - 487. i)T + (-7.38e5 + 5.36e5i)T^{2} \) |
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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
−15.72247465200859589488175794609, −13.98218544725243891038183987097, −12.98319917232831889188461865073, −11.70160843558563452243629960628, −11.12789232851320970177148576682, −9.241254153773777665987310335008, −8.212379889545375529000010155607, −6.08947314368302168211237129683, −3.09161522860825004543802797328, −1.27365000699255690642011968822,
4.30484362553940406749869714992, 5.98537108869656324299139203625, 7.61604330282450838736385765018, 8.864637709204605279993527656374, 10.04781453960966058139761978241, 11.57627276028000046287031667971, 13.86997238062127981630762514217, 14.69978052998909267493440659968, 15.72657232420817211876392014102, 16.63297509739926914204835945372
