| L(s) = 1 | + (2.50 − 2.98i)2-s + (−3.27 − 9.00i)3-s + (0.141 + 0.803i)4-s + (2.69 − 15.2i)5-s + (−35.0 − 12.7i)6-s + (18.9 + 32.8i)7-s + (56.7 + 32.7i)8-s + (−8.26 + 6.93i)9-s + (−38.9 − 46.3i)10-s + (−0.120 + 0.208i)11-s + (6.77 − 3.90i)12-s + (−91.3 + 251. i)13-s + (145. + 25.6i)14-s + (−146. + 25.8i)15-s + (227. − 82.8i)16-s + (−139. − 117. i)17-s + ⋯ |
| L(s) = 1 | + (0.626 − 0.746i)2-s + (−0.364 − 1.00i)3-s + (0.00885 + 0.0502i)4-s + (0.107 − 0.611i)5-s + (−0.974 − 0.354i)6-s + (0.386 + 0.669i)7-s + (0.886 + 0.511i)8-s + (−0.102 + 0.0856i)9-s + (−0.389 − 0.463i)10-s + (−0.000994 + 0.00172i)11-s + (0.0470 − 0.0271i)12-s + (−0.540 + 1.48i)13-s + (0.742 + 0.130i)14-s + (−0.651 + 0.114i)15-s + (0.889 − 0.323i)16-s + (−0.483 − 0.405i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.112 + 0.993i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.112 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.17827 - 1.05257i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.17827 - 1.05257i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + (261. + 248. i)T \) |
| good | 2 | \( 1 + (-2.50 + 2.98i)T + (-2.77 - 15.7i)T^{2} \) |
| 3 | \( 1 + (3.27 + 9.00i)T + (-62.0 + 52.0i)T^{2} \) |
| 5 | \( 1 + (-2.69 + 15.2i)T + (-587. - 213. i)T^{2} \) |
| 7 | \( 1 + (-18.9 - 32.8i)T + (-1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 + (0.120 - 0.208i)T + (-7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 + (91.3 - 251. i)T + (-2.18e4 - 1.83e4i)T^{2} \) |
| 17 | \( 1 + (139. + 117. i)T + (1.45e4 + 8.22e4i)T^{2} \) |
| 23 | \( 1 + (52.2 + 296. i)T + (-2.62e5 + 9.57e4i)T^{2} \) |
| 29 | \( 1 + (-560. - 667. i)T + (-1.22e5 + 6.96e5i)T^{2} \) |
| 31 | \( 1 + (1.34e3 - 773. i)T + (4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 + 550. iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (-907. - 2.49e3i)T + (-2.16e6 + 1.81e6i)T^{2} \) |
| 43 | \( 1 + (-462. + 2.62e3i)T + (-3.21e6 - 1.16e6i)T^{2} \) |
| 47 | \( 1 + (400. - 335. i)T + (8.47e5 - 4.80e6i)T^{2} \) |
| 53 | \( 1 + (1.61e3 - 284. i)T + (7.41e6 - 2.69e6i)T^{2} \) |
| 59 | \( 1 + (-4.37e3 + 5.21e3i)T + (-2.10e6 - 1.19e7i)T^{2} \) |
| 61 | \( 1 + (412. + 2.33e3i)T + (-1.30e7 + 4.73e6i)T^{2} \) |
| 67 | \( 1 + (2.26e3 + 2.70e3i)T + (-3.49e6 + 1.98e7i)T^{2} \) |
| 71 | \( 1 + (3.45e3 + 610. i)T + (2.38e7 + 8.69e6i)T^{2} \) |
| 73 | \( 1 + (5.30e3 - 1.93e3i)T + (2.17e7 - 1.82e7i)T^{2} \) |
| 79 | \( 1 + (-2.01e3 - 5.52e3i)T + (-2.98e7 + 2.50e7i)T^{2} \) |
| 83 | \( 1 + (1.76e3 + 3.04e3i)T + (-2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 + (-4.44e3 + 1.22e4i)T + (-4.80e7 - 4.03e7i)T^{2} \) |
| 97 | \( 1 + (6.47e3 - 7.71e3i)T + (-1.53e7 - 8.71e7i)T^{2} \) |
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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
−17.58873702506178889903946137046, −16.39317013462350589230897466870, −14.35556821265123120777184308410, −12.97791299513151905270653022202, −12.26313638818445685373506556142, −11.26123300570996288084537866863, −8.845711867080310959430195350665, −6.96508083040455451981509721516, −4.76662563396857121341651179849, −1.96913621350241948010152311909,
4.26704341469427673428921078981, 5.74941601672154793761458169079, 7.52283188657240377159226231601, 10.16936711600797442536026837931, 10.78508133407043389869698185266, 13.12268166902988995321497506487, 14.59443940755872989668218349527, 15.27785905382655453163573592325, 16.46151261806706780357308777943, 17.61026145957971809422539213779
