| L(s) = 1 | + (0.900 − 1.26i)2-s + (0.235 − 0.971i)3-s + (−0.134 − 0.387i)4-s + (−0.164 + 3.44i)5-s + (−1.01 − 1.17i)6-s + (−1.20 + 2.35i)7-s + (2.36 + 0.695i)8-s + (−0.888 − 0.458i)9-s + (4.20 + 3.30i)10-s + (−0.730 − 1.02i)11-s + (−0.408 + 0.0389i)12-s + (−0.0342 + 0.238i)13-s + (1.88 + 3.64i)14-s + (3.30 + 0.971i)15-s + (3.65 − 2.87i)16-s + (4.75 − 0.915i)17-s + ⋯ |
| L(s) = 1 | + (0.636 − 0.894i)2-s + (0.136 − 0.561i)3-s + (−0.0670 − 0.193i)4-s + (−0.0733 + 1.54i)5-s + (−0.415 − 0.478i)6-s + (−0.457 + 0.889i)7-s + (0.837 + 0.245i)8-s + (−0.296 − 0.152i)9-s + (1.33 + 1.04i)10-s + (−0.220 − 0.309i)11-s + (−0.117 + 0.0112i)12-s + (−0.00950 + 0.0660i)13-s + (0.504 + 0.975i)14-s + (0.854 + 0.250i)15-s + (0.914 − 0.718i)16-s + (1.15 − 0.222i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.100i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.06797 - 0.104577i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.06797 - 0.104577i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.235 + 0.971i)T \) |
| 7 | \( 1 + (1.20 - 2.35i)T \) |
| 23 | \( 1 + (4.29 - 2.13i)T \) |
| good | 2 | \( 1 + (-0.900 + 1.26i)T + (-0.654 - 1.89i)T^{2} \) |
| 5 | \( 1 + (0.164 - 3.44i)T + (-4.97 - 0.475i)T^{2} \) |
| 11 | \( 1 + (0.730 + 1.02i)T + (-3.59 + 10.3i)T^{2} \) |
| 13 | \( 1 + (0.0342 - 0.238i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (-4.75 + 0.915i)T + (15.7 - 6.31i)T^{2} \) |
| 19 | \( 1 + (-7.12 - 1.37i)T + (17.6 + 7.06i)T^{2} \) |
| 29 | \( 1 + (1.56 + 1.80i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (1.10 - 1.05i)T + (1.47 - 30.9i)T^{2} \) |
| 37 | \( 1 + (2.45 + 1.26i)T + (21.4 + 30.1i)T^{2} \) |
| 41 | \( 1 + (-5.95 - 3.82i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (0.169 - 0.0498i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + (5.18 + 8.97i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (5.90 + 2.36i)T + (38.3 + 36.5i)T^{2} \) |
| 59 | \( 1 + (10.5 + 8.28i)T + (13.9 + 57.3i)T^{2} \) |
| 61 | \( 1 + (-1.03 - 4.25i)T + (-54.2 + 27.9i)T^{2} \) |
| 67 | \( 1 + (1.12 + 0.107i)T + (65.7 + 12.6i)T^{2} \) |
| 71 | \( 1 + (5.24 + 11.4i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (-4.64 - 13.4i)T + (-57.3 + 45.1i)T^{2} \) |
| 79 | \( 1 + (-11.9 + 4.78i)T + (57.1 - 54.5i)T^{2} \) |
| 83 | \( 1 + (-15.1 + 9.76i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (11.3 + 10.8i)T + (4.23 + 88.8i)T^{2} \) |
| 97 | \( 1 + (-4.93 - 3.16i)T + (40.2 + 88.2i)T^{2} \) |
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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
−11.30965600916238459814871683324, −10.26854033444595037954618536669, −9.538687173214188258419734955098, −7.945646476398489913736711076678, −7.40916206852109960816964576016, −6.24143915949855154485080680286, −5.33800744692080445974854888473, −3.40610949200064628692589182311, −3.13171612145046522547750435744, −1.97353225558084123310050877745,
1.15129991346000896595240235057, 3.56563240256300213037594466597, 4.53979527810569936060797630373, 5.20866741523226850439855511450, 6.07892773838351427907940681103, 7.46979201711028372099207906771, 7.976040982213280062301581746308, 9.326142773611028020127802720609, 9.897576032773587801203756285852, 10.85582609550693852569775275538
