| L(s) = 1 | + 11.0·2-s + (−18.5 − 32.1i)3-s − 6.00·4-s + (52.2 + 90.4i)5-s + (−204. − 354. i)6-s + (−372. + 645. i)7-s − 1.48e3·8-s + (405. − 701. i)9-s + (576. + 999. i)10-s − 2.76e3·11-s + (111. + 192. i)12-s + (−5.34e3 + 9.25e3i)13-s + (−4.11e3 + 7.13e3i)14-s + (1.93e3 − 3.35e3i)15-s − 1.55e4·16-s + (−1.48e4 + 2.57e4i)17-s + ⋯ |
| L(s) = 1 | + 0.976·2-s + (−0.396 − 0.687i)3-s − 0.0469·4-s + (0.186 + 0.323i)5-s + (−0.387 − 0.670i)6-s + (−0.410 + 0.711i)7-s − 1.02·8-s + (0.185 − 0.320i)9-s + (0.182 + 0.315i)10-s − 0.626·11-s + (0.0186 + 0.0322i)12-s + (−0.674 + 1.16i)13-s + (−0.401 + 0.694i)14-s + (0.148 − 0.256i)15-s − 0.950·16-s + (−0.732 + 1.26i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.175518 + 0.470840i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.175518 + 0.470840i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (5.97e4 + 5.17e5i)T \) |
| good | 2 | \( 1 - 11.0T + 128T^{2} \) |
| 3 | \( 1 + (18.5 + 32.1i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 + (-52.2 - 90.4i)T + (-3.90e4 + 6.76e4i)T^{2} \) |
| 7 | \( 1 + (372. - 645. i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 + 2.76e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + (5.34e3 - 9.25e3i)T + (-3.13e7 - 5.43e7i)T^{2} \) |
| 17 | \( 1 + (1.48e4 - 2.57e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-1.67e4 - 2.89e4i)T + (-4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (5.15e4 + 8.92e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (-4.77e4 + 8.27e4i)T + (-8.62e9 - 1.49e10i)T^{2} \) |
| 31 | \( 1 + (-2.28e4 - 3.95e4i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + (4.55e4 + 7.89e4i)T + (-4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 + 8.24e5T + 1.94e11T^{2} \) |
| 47 | \( 1 - 1.28e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + (-1.52e5 - 2.63e5i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 + 8.50e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + (-4.19e4 + 7.27e4i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (6.02e5 + 1.04e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + (2.89e6 - 5.01e6i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 + (1.71e6 - 2.96e6i)T + (-5.52e12 - 9.56e12i)T^{2} \) |
| 79 | \( 1 + (1.08e6 - 1.87e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 + (1.63e6 + 2.83e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (-4.99e6 - 8.64e6i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 - 2.52e5T + 8.07e13T^{2} \) |
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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
−14.64841351507486326721939023530, −13.64764516053579135061909383197, −12.38564074088428909720670675590, −12.10726866361925399982709561531, −10.15102758838848317408679363654, −8.660950061587355013403117018116, −6.68754242347646894849564578947, −5.85135315834583760291102197924, −4.17496845658601367615440120403, −2.32450688725755853712368494824,
0.15240680272900231634654801640, 3.13691983696568160043995111947, 4.73383835045186979169648295795, 5.40418649493544558279129620214, 7.36376146078399918477216314191, 9.340613038592195042868721522279, 10.36750649338436091571940132652, 11.76178598412055537898808511890, 13.26551942113463372438850461579, 13.57189621837348653225053117488
