| L(s) = 1 | + (2.04 − 0.215i)3-s + (2.65 + 4.59i)5-s + (3.34 + 0.711i)7-s + (−4.65 + 0.990i)9-s + (7.11 + 6.40i)11-s + (4.90 − 11.0i)13-s + (6.41 + 8.83i)15-s + (0.726 − 0.654i)17-s + (8.63 − 3.84i)19-s + (7.00 + 0.735i)21-s + (−11.8 − 3.85i)23-s + (−1.56 + 2.70i)25-s + (−26.9 + 8.75i)27-s + (12.4 − 17.1i)29-s + (−24.6 + 18.7i)31-s + ⋯ |
| L(s) = 1 | + (0.682 − 0.0717i)3-s + (0.530 + 0.918i)5-s + (0.478 + 0.101i)7-s + (−0.517 + 0.110i)9-s + (0.647 + 0.582i)11-s + (0.377 − 0.846i)13-s + (0.427 + 0.588i)15-s + (0.0427 − 0.0385i)17-s + (0.454 − 0.202i)19-s + (0.333 + 0.0350i)21-s + (−0.515 − 0.167i)23-s + (−0.0624 + 0.108i)25-s + (−0.997 + 0.324i)27-s + (0.430 − 0.592i)29-s + (−0.795 + 0.605i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.903 - 0.429i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.903 - 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.80863 + 0.407930i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.80863 + 0.407930i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 31 | \( 1 + (24.6 - 18.7i)T \) |
| good | 3 | \( 1 + (-2.04 + 0.215i)T + (8.80 - 1.87i)T^{2} \) |
| 5 | \( 1 + (-2.65 - 4.59i)T + (-12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (-3.34 - 0.711i)T + (44.7 + 19.9i)T^{2} \) |
| 11 | \( 1 + (-7.11 - 6.40i)T + (12.6 + 120. i)T^{2} \) |
| 13 | \( 1 + (-4.90 + 11.0i)T + (-113. - 125. i)T^{2} \) |
| 17 | \( 1 + (-0.726 + 0.654i)T + (30.2 - 287. i)T^{2} \) |
| 19 | \( 1 + (-8.63 + 3.84i)T + (241. - 268. i)T^{2} \) |
| 23 | \( 1 + (11.8 + 3.85i)T + (427. + 310. i)T^{2} \) |
| 29 | \( 1 + (-12.4 + 17.1i)T + (-259. - 799. i)T^{2} \) |
| 37 | \( 1 + (47.5 + 27.4i)T + (684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-2.94 + 28.0i)T + (-1.64e3 - 349. i)T^{2} \) |
| 43 | \( 1 + (13.4 + 30.2i)T + (-1.23e3 + 1.37e3i)T^{2} \) |
| 47 | \( 1 + (31.0 - 22.5i)T + (682. - 2.10e3i)T^{2} \) |
| 53 | \( 1 + (4.38 + 20.6i)T + (-2.56e3 + 1.14e3i)T^{2} \) |
| 59 | \( 1 + (-6.11 - 58.1i)T + (-3.40e3 + 723. i)T^{2} \) |
| 61 | \( 1 + 49.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 + (-38.9 - 67.3i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + (-49.7 + 10.5i)T + (4.60e3 - 2.05e3i)T^{2} \) |
| 73 | \( 1 + (-22.9 - 20.6i)T + (557. + 5.29e3i)T^{2} \) |
| 79 | \( 1 + (37.4 - 33.7i)T + (652. - 6.20e3i)T^{2} \) |
| 83 | \( 1 + (-139. - 14.6i)T + (6.73e3 + 1.43e3i)T^{2} \) |
| 89 | \( 1 + (-162. + 52.9i)T + (6.40e3 - 4.65e3i)T^{2} \) |
| 97 | \( 1 + (1.47 + 4.53i)T + (-7.61e3 + 5.53e3i)T^{2} \) |
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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
−13.55168997723665168769001785817, −12.21766821157083646574011370943, −11.07278669169528831484828738319, −10.12246789774930484800748019021, −8.960202710975630773438942325898, −7.904865023153755646365346316708, −6.70705907804388547075141646103, −5.39710527287267880740627376748, −3.48978285442939353846204135865, −2.16135910609171227929594418106,
1.60511054200329065830760589075, 3.55388519553962523555898605254, 5.06057458285441283909725061268, 6.34491273274986905815366398992, 8.042431387448585168887569026098, 8.880384947817525897463670167573, 9.580377839289756342253083154114, 11.16395300391378168534991269994, 12.05265018365457910063155015622, 13.37267022174175173272951741167
