| L(s) = 1 | + (−2.44 − 1.73i)3-s + (4.5 − 2.59i)5-s + (−4.17 + 7.22i)7-s + (2.99 + 8.48i)9-s + (0.825 + 0.476i)11-s + (−4.84 − 8.39i)13-s + (−15.5 − 1.43i)15-s − 18.8i·17-s − 24.6·19-s + (22.7 − 10.4i)21-s + (−0.825 + 0.476i)23-s + (1 − 1.73i)25-s + (7.34 − 25.9i)27-s + (−11.8 − 6.84i)29-s + (1.52 + 2.63i)31-s + ⋯ |
| L(s) = 1 | + (−0.816 − 0.577i)3-s + (0.900 − 0.519i)5-s + (−0.596 + 1.03i)7-s + (0.333 + 0.942i)9-s + (0.0750 + 0.0433i)11-s + (−0.372 − 0.645i)13-s + (−1.03 − 0.0953i)15-s − 1.11i·17-s − 1.29·19-s + (1.08 − 0.499i)21-s + (−0.0359 + 0.0207i)23-s + (0.0400 − 0.0692i)25-s + (0.272 − 0.962i)27-s + (−0.408 − 0.235i)29-s + (0.0491 + 0.0850i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.936 - 0.350i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.936 - 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.08585026330\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08585026330\) |
| \(L(2)\) |
|
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 | \( 1 + (4.17 - 7.22i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-0.825 - 0.476i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (4.84 + 8.39i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + 18.8iT - 289T^{2} \) |
| 19 | \( 1 + 24.6T + 361T^{2} \) |
| 23 | \( 1 + (0.825 - 0.476i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (11.8 + 6.84i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-1.52 - 2.63i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 46.6T + 1.36e3T^{2} \) |
| 41 | \( 1 + (9.45 - 5.45i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (22.5 - 39.0i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (39.2 + 22.6i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 - 94.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (16.2 - 9.39i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-6.54 + 11.3i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (37.5 + 64.9i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 18.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 7.90T + 5.32e3T^{2} \) |
| 79 | \( 1 + (21.8 - 37.8i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (112. + 65.1i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 - 145. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-54.9 + 95.1i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−9.978250288863614250943042614776, −9.255762658809239901524647497874, −8.315333958530969617748960974804, −7.12655971716861011197954243094, −6.18174945966648314597604852453, −5.55500507835713216341604628577, −4.74302383927982544278648517969, −2.79502802025449987847132248846, −1.71357653967474329733869377495, −0.03397242950084387879984588801,
1.83076158395789606916358366616, 3.54244097487525556179003260428, 4.38182885062435956841225093432, 5.62391994042219235850222016039, 6.55067386363148248894577142827, 6.94023902578078002041199752490, 8.539229973247792457739783881611, 9.619869152809187347491996460022, 10.31115901567874109409971776897, 10.64078180336186869330717090889
