| L(s) = 1 | + (−1.94 − 0.485i)2-s + (2.89 − 0.777i)3-s + (3.52 + 1.88i)4-s + (0.130 + 0.739i)5-s + (−5.99 + 0.101i)6-s + (−3.97 − 3.33i)7-s + (−5.93 − 5.36i)8-s + (7.79 − 4.50i)9-s + (0.106 − 1.49i)10-s + (0.771 − 4.37i)11-s + (11.6 + 2.71i)12-s + (8.39 − 23.0i)13-s + (6.09 + 8.39i)14-s + (0.953 + 2.04i)15-s + (8.90 + 13.2i)16-s + (−14.9 + 8.60i)17-s + ⋯ |
| L(s) = 1 | + (−0.970 − 0.242i)2-s + (0.965 − 0.259i)3-s + (0.882 + 0.470i)4-s + (0.0260 + 0.147i)5-s + (−0.999 + 0.0169i)6-s + (−0.567 − 0.476i)7-s + (−0.741 − 0.670i)8-s + (0.865 − 0.500i)9-s + (0.0106 − 0.149i)10-s + (0.0701 − 0.397i)11-s + (0.974 + 0.226i)12-s + (0.645 − 1.77i)13-s + (0.435 + 0.599i)14-s + (0.0635 + 0.136i)15-s + (0.556 + 0.830i)16-s + (−0.876 + 0.506i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.335 + 0.942i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.335 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.09206 - 0.770589i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.09206 - 0.770589i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.94 + 0.485i)T \) |
| 3 | \( 1 + (-2.89 + 0.777i)T \) |
| good | 5 | \( 1 + (-0.130 - 0.739i)T + (-23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (3.97 + 3.33i)T + (8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (-0.771 + 4.37i)T + (-113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (-8.39 + 23.0i)T + (-129. - 108. i)T^{2} \) |
| 17 | \( 1 + (14.9 - 8.60i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-29.6 - 17.1i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (14.1 + 16.8i)T + (-91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (-13.9 + 5.06i)T + (644. - 540. i)T^{2} \) |
| 31 | \( 1 + (-15.8 + 13.3i)T + (166. - 946. i)T^{2} \) |
| 37 | \( 1 + (41.1 - 23.7i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-13.1 + 36.2i)T + (-1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (22.1 + 3.91i)T + (1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (33.6 - 40.1i)T + (-383. - 2.17e3i)T^{2} \) |
| 53 | \( 1 - 48.8T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-14.9 - 84.7i)T + (-3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (-12.4 + 14.8i)T + (-646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (34.5 - 94.8i)T + (-3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (98.8 - 57.0i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (11.4 - 19.7i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (25.1 - 9.13i)T + (4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (-71.6 + 26.0i)T + (5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (12.8 + 7.43i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (4.65 - 26.3i)T + (-8.84e3 - 3.21e3i)T^{2} \) |
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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
−11.88592892750340846195273157879, −10.37391308260827939774623201325, −10.15279171529744002267508082255, −8.750189341545624661210395323855, −8.149404393393097942452623997151, −7.15041883691685611163708584960, −6.07615066880669940273977734503, −3.67772000210617335662829068162, −2.80225408010714207838233638297, −0.980631198600104849726020450895,
1.76113911158511988301092151374, 3.14062912830906503488982609042, 4.86470830406939595070497296859, 6.58756637176339107077873441515, 7.29869502196070632650556604784, 8.699102853746251058168990334928, 9.194852897917650703372186787014, 9.849742446319635770769567469810, 11.18509478329170336406638975808, 12.04438966864193668573511373575
