| L(s) = 1 | + (−3.49 + 4.03i)3-s + (−12.4 + 86.8i)5-s + (−77.3 − 169. i)7-s + (30.5 + 212. i)9-s + (−341. + 100. i)11-s + (260. − 569. i)13-s + (−306. − 354. i)15-s + (1.66e3 − 1.06e3i)17-s + (−1.79e3 − 1.15e3i)19-s + (953. + 280. i)21-s + (−2.48e3 + 532. i)23-s + (−4.39e3 − 1.29e3i)25-s + (−2.05e3 − 1.31e3i)27-s + (−1.14e3 + 733. i)29-s + (−4.54e3 − 5.25e3i)31-s + ⋯ |
| L(s) = 1 | + (−0.224 + 0.258i)3-s + (−0.223 + 1.55i)5-s + (−0.596 − 1.30i)7-s + (0.125 + 0.873i)9-s + (−0.851 + 0.249i)11-s + (0.427 − 0.935i)13-s + (−0.352 − 0.406i)15-s + (1.39 − 0.897i)17-s + (−1.14 − 0.734i)19-s + (0.471 + 0.138i)21-s + (−0.977 + 0.209i)23-s + (−1.40 − 0.413i)25-s + (−0.542 − 0.348i)27-s + (−0.251 + 0.161i)29-s + (−0.850 − 0.981i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.583 + 0.812i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.583 + 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.0980892 - 0.191116i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0980892 - 0.191116i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 + (2.48e3 - 532. i)T \) |
| good | 3 | \( 1 + (3.49 - 4.03i)T + (-34.5 - 240. i)T^{2} \) |
| 5 | \( 1 + (12.4 - 86.8i)T + (-2.99e3 - 880. i)T^{2} \) |
| 7 | \( 1 + (77.3 + 169. i)T + (-1.10e4 + 1.27e4i)T^{2} \) |
| 11 | \( 1 + (341. - 100. i)T + (1.35e5 - 8.70e4i)T^{2} \) |
| 13 | \( 1 + (-260. + 569. i)T + (-2.43e5 - 2.80e5i)T^{2} \) |
| 17 | \( 1 + (-1.66e3 + 1.06e3i)T + (5.89e5 - 1.29e6i)T^{2} \) |
| 19 | \( 1 + (1.79e3 + 1.15e3i)T + (1.02e6 + 2.25e6i)T^{2} \) |
| 29 | \( 1 + (1.14e3 - 733. i)T + (8.52e6 - 1.86e7i)T^{2} \) |
| 31 | \( 1 + (4.54e3 + 5.25e3i)T + (-4.07e6 + 2.83e7i)T^{2} \) |
| 37 | \( 1 + (1.69e3 + 1.18e4i)T + (-6.65e7 + 1.95e7i)T^{2} \) |
| 41 | \( 1 + (1.36e3 - 9.47e3i)T + (-1.11e8 - 3.26e7i)T^{2} \) |
| 43 | \( 1 + (-1.78e3 + 2.05e3i)T + (-2.09e7 - 1.45e8i)T^{2} \) |
| 47 | \( 1 + 2.52e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + (-7.59e3 - 1.66e4i)T + (-2.73e8 + 3.16e8i)T^{2} \) |
| 59 | \( 1 + (8.39e3 - 1.83e4i)T + (-4.68e8 - 5.40e8i)T^{2} \) |
| 61 | \( 1 + (-2.02e4 - 2.33e4i)T + (-1.20e8 + 8.35e8i)T^{2} \) |
| 67 | \( 1 + (-6.47e4 - 1.90e4i)T + (1.13e9 + 7.29e8i)T^{2} \) |
| 71 | \( 1 + (5.71e4 + 1.67e4i)T + (1.51e9 + 9.75e8i)T^{2} \) |
| 73 | \( 1 + (6.82e4 + 4.38e4i)T + (8.61e8 + 1.88e9i)T^{2} \) |
| 79 | \( 1 + (3.39e4 - 7.44e4i)T + (-2.01e9 - 2.32e9i)T^{2} \) |
| 83 | \( 1 + (1.34e3 + 9.32e3i)T + (-3.77e9 + 1.10e9i)T^{2} \) |
| 89 | \( 1 + (2.45e3 - 2.83e3i)T + (-7.94e8 - 5.52e9i)T^{2} \) |
| 97 | \( 1 + (586. - 4.08e3i)T + (-8.23e9 - 2.41e9i)T^{2} \) |
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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
−12.95347032972236624181934641136, −11.22434891676474053598539702895, −10.48878644164788898445466335521, −10.02738933667835830825465142751, −7.72964380813299414117373820351, −7.21559759198490631040393957364, −5.71653053824520360171200634251, −3.96239014093520708439183137389, −2.71093686263478497429868535632, −0.086885101840385521981645747357,
1.62884879599496011199679969607, 3.76456460125041633949977201117, 5.40305442811258417981378240387, 6.26068756519420728249382706414, 8.228239483895275777231970297586, 8.872936633517850915043977565247, 10.00758252263983240412575883211, 11.82505688322042916678818431102, 12.46296663636994886723186180127, 12.95848685928459114526641019960
