| L(s) = 1 | + (−0.762 − 2.34i)2-s + (−1.28 + 1.15i)3-s + (−1.68 + 1.22i)4-s + (4.41 + 7.65i)5-s + (3.69 + 2.13i)6-s + (1.12 + 10.6i)7-s + (−3.82 − 2.77i)8-s + (0.313 − 2.98i)9-s + (14.5 − 16.1i)10-s + (5.66 − 12.7i)11-s + (0.750 − 3.53i)12-s + (2.54 + 11.9i)13-s + (24.2 − 10.7i)14-s + (−14.5 − 4.72i)15-s + (−6.17 + 19.0i)16-s + (−0.627 − 1.40i)17-s + ⋯ |
| L(s) = 1 | + (−0.381 − 1.17i)2-s + (−0.429 + 0.386i)3-s + (−0.421 + 0.306i)4-s + (0.883 + 1.53i)5-s + (0.616 + 0.356i)6-s + (0.160 + 1.52i)7-s + (−0.477 − 0.347i)8-s + (0.0348 − 0.331i)9-s + (1.45 − 1.61i)10-s + (0.514 − 1.15i)11-s + (0.0625 − 0.294i)12-s + (0.195 + 0.920i)13-s + (1.72 − 0.769i)14-s + (−0.970 − 0.315i)15-s + (−0.386 + 1.18i)16-s + (−0.0368 − 0.0828i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0656i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.997 - 0.0656i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.09410 + 0.0359621i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.09410 + 0.0359621i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.28 - 1.15i)T \) |
| 31 | \( 1 + (-23.4 + 20.3i)T \) |
| good | 2 | \( 1 + (0.762 + 2.34i)T + (-3.23 + 2.35i)T^{2} \) |
| 5 | \( 1 + (-4.41 - 7.65i)T + (-12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (-1.12 - 10.6i)T + (-47.9 + 10.1i)T^{2} \) |
| 11 | \( 1 + (-5.66 + 12.7i)T + (-80.9 - 89.9i)T^{2} \) |
| 13 | \( 1 + (-2.54 - 11.9i)T + (-154. + 68.7i)T^{2} \) |
| 17 | \( 1 + (0.627 + 1.40i)T + (-193. + 214. i)T^{2} \) |
| 19 | \( 1 + (-8.80 - 1.87i)T + (329. + 146. i)T^{2} \) |
| 23 | \( 1 + (-11.9 + 16.4i)T + (-163. - 503. i)T^{2} \) |
| 29 | \( 1 + (26.1 - 8.49i)T + (680. - 494. i)T^{2} \) |
| 37 | \( 1 + (8.41 + 4.86i)T + (684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-13.7 + 15.2i)T + (-175. - 1.67e3i)T^{2} \) |
| 43 | \( 1 + (-11.3 + 53.6i)T + (-1.68e3 - 752. i)T^{2} \) |
| 47 | \( 1 + (12.8 - 39.5i)T + (-1.78e3 - 1.29e3i)T^{2} \) |
| 53 | \( 1 + (-54.8 - 5.76i)T + (2.74e3 + 584. i)T^{2} \) |
| 59 | \( 1 + (62.6 + 69.6i)T + (-363. + 3.46e3i)T^{2} \) |
| 61 | \( 1 + 64.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 + (26.5 + 45.9i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + (-10.1 + 96.3i)T + (-4.93e3 - 1.04e3i)T^{2} \) |
| 73 | \( 1 + (28.0 - 63.0i)T + (-3.56e3 - 3.96e3i)T^{2} \) |
| 79 | \( 1 + (-24.2 - 54.4i)T + (-4.17e3 + 4.63e3i)T^{2} \) |
| 83 | \( 1 + (-90.0 - 81.1i)T + (720. + 6.85e3i)T^{2} \) |
| 89 | \( 1 + (54.9 + 75.6i)T + (-2.44e3 + 7.53e3i)T^{2} \) |
| 97 | \( 1 + (-31.6 + 22.9i)T + (2.90e3 - 8.94e3i)T^{2} \) |
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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
−13.80910811935810089350174241916, −12.23554260469940591223386708717, −11.31505045397316185717282092353, −10.85781587328381201040866612505, −9.612594540195128302274558341739, −8.937155135696680551940105578691, −6.53517492929998460193708221616, −5.79530750884230346782010895274, −3.31410463342548343020309496309, −2.16442011320007725251012134496,
1.14937412210628412225426135295, 4.65669762939831296403657736864, 5.73369162667944478738148410708, 7.04085033753232153357139871879, 7.909263649336316022141427820663, 9.182155547374926509246208363842, 10.20319655082286992097912261574, 11.85328353161558684300571457926, 13.01028013073605154428637797511, 13.68294270284662697705802960348
