| L(s) = 1 | + (−0.959 + 0.281i)2-s + (0.712 + 0.822i)3-s + (0.841 − 0.540i)4-s + (0.174 + 1.21i)5-s + (−0.915 − 0.588i)6-s + (0.260 − 0.570i)7-s + (−0.654 + 0.755i)8-s + (0.258 − 1.79i)9-s + (−0.510 − 1.11i)10-s + (−5.65 − 1.65i)11-s + (1.04 + 0.306i)12-s + (−0.977 − 2.13i)13-s + (−0.0892 + 0.620i)14-s + (−0.875 + 1.01i)15-s + (0.415 − 0.909i)16-s + (4.49 + 2.88i)17-s + ⋯ |
| L(s) = 1 | + (−0.678 + 0.199i)2-s + (0.411 + 0.474i)3-s + (0.420 − 0.270i)4-s + (0.0782 + 0.543i)5-s + (−0.373 − 0.240i)6-s + (0.0984 − 0.215i)7-s + (−0.231 + 0.267i)8-s + (0.0861 − 0.599i)9-s + (−0.161 − 0.353i)10-s + (−1.70 − 0.500i)11-s + (0.301 + 0.0884i)12-s + (−0.270 − 0.593i)13-s + (−0.0238 + 0.165i)14-s + (−0.226 + 0.260i)15-s + (0.103 − 0.227i)16-s + (1.08 + 0.700i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 - 0.540i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.841 - 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.647203 + 0.189887i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.647203 + 0.189887i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.959 - 0.281i)T \) |
| 23 | \( 1 + (2.50 - 4.09i)T \) |
| good | 3 | \( 1 + (-0.712 - 0.822i)T + (-0.426 + 2.96i)T^{2} \) |
| 5 | \( 1 + (-0.174 - 1.21i)T + (-4.79 + 1.40i)T^{2} \) |
| 7 | \( 1 + (-0.260 + 0.570i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (5.65 + 1.65i)T + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (0.977 + 2.13i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (-4.49 - 2.88i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (0.227 - 0.146i)T + (7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (1.66 + 1.07i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (4.32 - 4.99i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (1.43 - 9.96i)T + (-35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (1.53 + 10.7i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (3.01 + 3.47i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 - 6.97T + 47T^{2} \) |
| 53 | \( 1 + (1.09 - 2.39i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (-0.685 - 1.50i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (5.98 - 6.90i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (-5.94 + 1.74i)T + (56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (-8.69 + 2.55i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (4.99 - 3.21i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (2.49 + 5.46i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-0.277 + 1.93i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (3.00 + 3.46i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (1.44 + 10.0i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−15.75107344534509281543096913678, −15.09637314139661780221254791308, −13.91602269220767647937391001788, −12.37749057578340317876800204449, −10.63596554257395446937554317472, −10.08116643459923104997412530132, −8.536494501746372259305347970728, −7.40048260652854408068734902981, −5.62372989618455292652605235635, −3.17520889943630460572375792586,
2.35454866121904277363798590123, 5.14972666621727978338394967570, 7.36333181976043825519698709994, 8.218220041083991670732780650561, 9.596742883310064667267237508555, 10.81664254232621625136172098390, 12.36412474115584037294281041317, 13.17945408179136672855031906898, 14.55973669097802515884743349136, 16.03994756686736293139096617402
