| L(s) = 1 | + (0.766 − 0.642i)2-s + (−0.764 + 1.55i)3-s + (0.173 − 0.984i)4-s + (3.01 − 1.09i)5-s + (0.412 + 1.68i)6-s + (−0.173 − 0.984i)7-s + (−0.500 − 0.866i)8-s + (−1.82 − 2.37i)9-s + (1.60 − 2.78i)10-s + (2.50 + 0.911i)11-s + (1.39 + 1.02i)12-s + (−0.986 − 0.827i)13-s + (−0.766 − 0.642i)14-s + (−0.601 + 5.53i)15-s + (−0.939 − 0.342i)16-s + (2.53 − 4.38i)17-s + ⋯ |
| L(s) = 1 | + (0.541 − 0.454i)2-s + (−0.441 + 0.897i)3-s + (0.0868 − 0.492i)4-s + (1.34 − 0.491i)5-s + (0.168 + 0.686i)6-s + (−0.0656 − 0.372i)7-s + (−0.176 − 0.306i)8-s + (−0.609 − 0.792i)9-s + (0.507 − 0.879i)10-s + (0.754 + 0.274i)11-s + (0.403 + 0.295i)12-s + (−0.273 − 0.229i)13-s + (−0.204 − 0.171i)14-s + (−0.155 + 1.42i)15-s + (−0.234 − 0.0855i)16-s + (0.614 − 1.06i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 + 0.427i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.904 + 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.87396 - 0.420448i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.87396 - 0.420448i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.766 + 0.642i)T \) |
| 3 | \( 1 + (0.764 - 1.55i)T \) |
| 7 | \( 1 + (0.173 + 0.984i)T \) |
| good | 5 | \( 1 + (-3.01 + 1.09i)T + (3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 + (-2.50 - 0.911i)T + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (0.986 + 0.827i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.53 + 4.38i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.18 - 5.51i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.892 - 5.06i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (6.01 - 5.04i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-1.43 + 8.13i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-0.199 + 0.346i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.28 + 1.07i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-1.66 - 0.605i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (0.231 + 1.31i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + 6.04T + 53T^{2} \) |
| 59 | \( 1 + (14.3 - 5.21i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.94 - 11.0i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (0.944 + 0.792i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (4.29 - 7.43i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (5.30 + 9.19i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (7.99 - 6.70i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (8.11 - 6.80i)T + (14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (6.48 + 11.2i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.63 - 0.595i)T + (74.3 + 62.3i)T^{2} \) |
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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
−11.38004593338386090128334493172, −10.18571269844578477590359183109, −9.700073610218188677799504165967, −9.152219705144660774820209581923, −7.37126405728425142232536697491, −5.88935445856243935325110229209, −5.51281078592723849744804116154, −4.37670373097683178819422584519, −3.21619490262269760632548461654, −1.45949582585226913549893362114,
1.79068287834924222511967729499, 3.00878859650219445821936412301, 4.88931962238527264697347548165, 5.97902443744956368544502208442, 6.37229277197900741207597689219, 7.31766232816800140722831016274, 8.536875844320637712583910186877, 9.543577483394046173950041312478, 10.68741619245637609447844517229, 11.60423357354952673172306092044
