| L(s) = 1 | + (−51.8 + 62.2i)3-s + (−254. − 570. i)5-s − 4.20e3i·7-s + (−1.17e3 − 6.45e3i)9-s + 542. i·11-s − 4.07e4i·13-s + (4.87e4 + 1.37e4i)15-s + 3.12e4·17-s − 2.30e5·19-s + (2.61e5 + 2.18e5i)21-s + 4.65e5·23-s + (−2.60e5 + 2.90e5i)25-s + (4.62e5 + 2.61e5i)27-s + 2.31e5i·29-s − 2.06e5·31-s + ⋯ |
| L(s) = 1 | + (−0.640 + 0.767i)3-s + (−0.407 − 0.913i)5-s − 1.75i·7-s + (−0.179 − 0.983i)9-s + 0.0370i·11-s − 1.42i·13-s + (0.962 + 0.271i)15-s + 0.373·17-s − 1.77·19-s + (1.34 + 1.12i)21-s + 1.66·23-s + (−0.667 + 0.744i)25-s + (0.870 + 0.492i)27-s + 0.326i·29-s − 0.223·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 - 0.271i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.962 - 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.6203472693\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6203472693\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (51.8 - 62.2i)T \) |
| 5 | \( 1 + (254. + 570. i)T \) |
| good | 7 | \( 1 + 4.20e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 - 542. iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 4.07e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 - 3.12e4T + 6.97e9T^{2} \) |
| 19 | \( 1 + 2.30e5T + 1.69e10T^{2} \) |
| 23 | \( 1 - 4.65e5T + 7.83e10T^{2} \) |
| 29 | \( 1 - 2.31e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 2.06e5T + 8.52e11T^{2} \) |
| 37 | \( 1 + 2.73e6iT - 3.51e12T^{2} \) |
| 41 | \( 1 + 2.74e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 6.38e5iT - 1.16e13T^{2} \) |
| 47 | \( 1 - 2.05e6T + 2.38e13T^{2} \) |
| 53 | \( 1 + 1.09e7T + 6.22e13T^{2} \) |
| 59 | \( 1 + 5.20e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 7.93e3T + 1.91e14T^{2} \) |
| 67 | \( 1 + 2.61e7iT - 4.06e14T^{2} \) |
| 71 | \( 1 - 4.61e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 3.27e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 - 9.49e6T + 1.51e15T^{2} \) |
| 83 | \( 1 - 6.39e7T + 2.25e15T^{2} \) |
| 89 | \( 1 - 4.56e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 2.57e7iT - 7.83e15T^{2} \) |
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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
−10.90127973311782726070798455618, −10.63758489106177445129727076502, −9.356071599560880957682232624015, −8.120966550953641073781151393917, −6.94334497486081352203199000244, −5.42296382141315652270123781074, −4.41738041359241387871916580251, −3.54224502986619345670494436706, −0.952413013785103944671677767986, −0.22477932365163830049091167028,
1.81676363958034331769516480161, 2.80904161202622234157128145086, 4.74986353278260292379117474786, 6.15617650395003458182631943936, 6.74091640204457073424817177686, 8.126301161690422711020025512551, 9.165100923764972329498710678725, 10.75966765399750052352675092730, 11.59694221595637534844184189230, 12.21606708456217315498147349502
