| L(s) = 1 | + (−2.44 − 1.73i)3-s + (4.5 + 2.59i)5-s + (−3.17 − 5.49i)7-s + (2.99 + 8.48i)9-s + (−8.17 + 4.71i)11-s + (9.84 − 17.0i)13-s + (−6.52 − 14.1i)15-s + 1.90i·17-s − 4.69·19-s + (−1.74 + 18.9i)21-s + (8.17 + 4.71i)23-s + (1 + 1.73i)25-s + (7.34 − 25.9i)27-s + (2.84 − 1.64i)29-s + (20.5 − 35.5i)31-s + ⋯ |
| L(s) = 1 | + (−0.816 − 0.577i)3-s + (0.900 + 0.519i)5-s + (−0.453 − 0.785i)7-s + (0.333 + 0.942i)9-s + (−0.743 + 0.429i)11-s + (0.757 − 1.31i)13-s + (−0.434 − 0.943i)15-s + 0.112i·17-s − 0.247·19-s + (−0.0832 + 0.903i)21-s + (0.355 + 0.205i)23-s + (0.0400 + 0.0692i)25-s + (0.272 − 0.962i)27-s + (0.0982 − 0.0567i)29-s + (0.662 − 1.14i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.507 + 0.861i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.507 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.018208163\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.018208163\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.44 + 1.73i)T \) |
| good | 5 | \( 1 + (-4.5 - 2.59i)T + (12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (3.17 + 5.49i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (8.17 - 4.71i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-9.84 + 17.0i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 - 1.90iT - 289T^{2} \) |
| 19 | \( 1 + 4.69T + 361T^{2} \) |
| 23 | \( 1 + (-8.17 - 4.71i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-2.84 + 1.64i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (-20.5 + 35.5i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + 17.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + (53.5 + 30.9i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-0.477 - 0.826i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (12.2 - 7.05i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + 9.53iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (79.2 + 45.7i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (37.5 + 65.0i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-15.4 + 26.8i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 85.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 96.0T + 5.32e3T^{2} \) |
| 79 | \( 1 + (14.8 + 25.7i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-76.1 + 43.9i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 - 41.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (47.9 + 83.0i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−10.50076572432282002059890003803, −9.716704097527234547233011637823, −8.198455565898479357038789177436, −7.40345351027974449038669431570, −6.45675547300990486140917428583, −5.83761185605613546786644700295, −4.82266563592910423234680002795, −3.26563869002137803807200024278, −1.94684465710613009896830986851, −0.43513553535101963819459995794,
1.46259093000202735912153423796, 3.03139834381801708860938679630, 4.45903860941447660231829212994, 5.37810658272472006534594131804, 6.07522115225760080378282836565, 6.84654937061642981864999299892, 8.600718156711252120540742490114, 9.095567875194322863222654387585, 9.949506215139892655598965521414, 10.73264785873547654720081797365
