| L(s) = 1 | + (0.961 + 1.03i)2-s + (−1.03 + 1.39i)3-s + (−0.149 + 1.99i)4-s + (−0.337 + 0.926i)5-s + (−2.43 + 0.270i)6-s + (0.984 + 0.173i)7-s + (−2.21 + 1.76i)8-s + (−0.876 − 2.86i)9-s + (−1.28 + 0.541i)10-s + (−4.05 + 1.47i)11-s + (−2.62 − 2.26i)12-s + (1.89 − 1.59i)13-s + (0.767 + 1.18i)14-s + (−0.942 − 1.42i)15-s + (−3.95 − 0.596i)16-s + (−4.64 + 2.68i)17-s + ⋯ |
| L(s) = 1 | + (0.680 + 0.733i)2-s + (−0.594 + 0.803i)3-s + (−0.0747 + 0.997i)4-s + (−0.150 + 0.414i)5-s + (−0.993 + 0.110i)6-s + (0.372 + 0.0656i)7-s + (−0.781 + 0.623i)8-s + (−0.292 − 0.956i)9-s + (−0.406 + 0.171i)10-s + (−1.22 + 0.445i)11-s + (−0.757 − 0.653i)12-s + (0.526 − 0.441i)13-s + (0.205 + 0.317i)14-s + (−0.243 − 0.367i)15-s + (−0.988 − 0.149i)16-s + (−1.12 + 0.650i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.733 + 0.680i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.733 + 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.377444 - 0.961698i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.377444 - 0.961698i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.961 - 1.03i)T \) |
| 3 | \( 1 + (1.03 - 1.39i)T \) |
| 7 | \( 1 + (-0.984 - 0.173i)T \) |
| good | 5 | \( 1 + (0.337 - 0.926i)T + (-3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 + (4.05 - 1.47i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-1.89 + 1.59i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (4.64 - 2.68i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.40 + 1.96i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.28 - 7.31i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-5.80 + 6.91i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (0.573 - 0.101i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (0.657 + 1.13i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (6.21 + 7.40i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.107 - 0.295i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (1.36 - 7.74i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 4.31iT - 53T^{2} \) |
| 59 | \( 1 + (5.36 + 1.95i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (0.255 - 1.44i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-2.87 - 3.42i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-8.25 - 14.2i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (3.58 - 6.20i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5.03 - 6.00i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (5.55 + 4.65i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-12.4 - 7.16i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (4.61 - 1.68i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.99185772647828825573619728405, −10.20834452291254125300766864814, −9.000333050155555828265476340350, −8.222912673851848094834469928295, −7.22913536881352407677286637609, −6.32196201206732993608622959034, −5.45535496706964319559432410885, −4.69223628751776428659212448144, −3.79367886106618654681076872440, −2.63084660241829564992183629160,
0.43252746913348836045999482340, 1.87201231442922078308976208781, 2.94042222571929868802448287826, 4.64997849975687200014944930176, 4.98040418068211135598163448117, 6.25537058475303410384682048435, 6.83227843261696177540124232710, 8.284460728383687120007793179188, 8.775060044206806275744540501924, 10.40424595223353818418935092962
