| L(s) = 1 | + (0.483 − 1.32i)2-s + (0.320 + 2.98i)3-s + (−1.53 − 1.28i)4-s + (5.90 + 1.04i)5-s + (4.11 + 1.01i)6-s + (5.59 − 4.69i)7-s + (−2.44 + 1.41i)8-s + (−8.79 + 1.91i)9-s + (4.23 − 7.34i)10-s + (−20.3 + 3.59i)11-s + (3.34 − 4.98i)12-s + (6.40 − 2.33i)13-s + (−3.53 − 9.71i)14-s + (−1.20 + 17.9i)15-s + (0.694 + 3.93i)16-s + (1.96 + 1.13i)17-s + ⋯ |
| L(s) = 1 | + (0.241 − 0.664i)2-s + (0.106 + 0.994i)3-s + (−0.383 − 0.321i)4-s + (1.18 + 0.208i)5-s + (0.686 + 0.169i)6-s + (0.799 − 0.671i)7-s + (−0.306 + 0.176i)8-s + (−0.977 + 0.212i)9-s + (0.423 − 0.734i)10-s + (−1.85 + 0.326i)11-s + (0.278 − 0.415i)12-s + (0.492 − 0.179i)13-s + (−0.252 − 0.693i)14-s + (−0.0806 + 1.19i)15-s + (0.0434 + 0.246i)16-s + (0.115 + 0.0667i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.143i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.989 + 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.42603 - 0.102725i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.42603 - 0.102725i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.483 + 1.32i)T \) |
| 3 | \( 1 + (-0.320 - 2.98i)T \) |
| good | 5 | \( 1 + (-5.90 - 1.04i)T + (23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (-5.59 + 4.69i)T + (8.50 - 48.2i)T^{2} \) |
| 11 | \( 1 + (20.3 - 3.59i)T + (113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (-6.40 + 2.33i)T + (129. - 108. i)T^{2} \) |
| 17 | \( 1 + (-1.96 - 1.13i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (12.1 + 21.0i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (15.5 - 18.5i)T + (-91.8 - 520. i)T^{2} \) |
| 29 | \( 1 + (-6.32 + 17.3i)T + (-644. - 540. i)T^{2} \) |
| 31 | \( 1 + (-16.6 - 13.9i)T + (166. + 946. i)T^{2} \) |
| 37 | \( 1 + (-22.3 + 38.7i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-17.5 - 48.1i)T + (-1.28e3 + 1.08e3i)T^{2} \) |
| 43 | \( 1 + (-6.41 - 36.3i)T + (-1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (-8.15 - 9.71i)T + (-383. + 2.17e3i)T^{2} \) |
| 53 | \( 1 + 17.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-64.9 - 11.4i)T + (3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (-32.3 + 27.1i)T + (646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (111. - 40.4i)T + (3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (46.1 + 26.6i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-33.8 - 58.6i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-104. - 38.0i)T + (4.78e3 + 4.01e3i)T^{2} \) |
| 83 | \( 1 + (-8.63 + 23.7i)T + (-5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (35.4 - 20.4i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-7.02 - 39.8i)T + (-8.84e3 + 3.21e3i)T^{2} \) |
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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
−14.92817190244522674129892465077, −13.82973466868233419891362729756, −13.14004942153981720913948077904, −11.18137352670377737396444068776, −10.46997567847221656283300427835, −9.646506546170726374302772556045, −8.076408782720419779323826776730, −5.70537321916241728598723569154, −4.55675581519798417268653461907, −2.54443760354670577729033765190,
2.26485979238134437549567390238, 5.34822171942748539850011891057, 6.14005312120654799924888898173, 7.895854629753285982683969700626, 8.673656772823837132699038281628, 10.44761319654681122853666920431, 12.13204730585545345598417985861, 13.17787487762470231260021246811, 13.87928894274968829674755140342, 14.91902165795005601438885060251
