| L(s) = 1 | + (−4.61 + 7.99i)5-s + (5.33 − 3.07i)7-s + (−3.70 + 2.13i)11-s + (0.869 − 1.50i)13-s − 12.3·17-s + 33.9i·19-s + (−3.35 − 1.93i)23-s + (−30.1 − 52.1i)25-s + (−17.8 − 30.9i)29-s + (−38.8 − 22.4i)31-s + 56.8i·35-s − 32.7·37-s + (−21.8 + 37.8i)41-s + (−33.9 + 19.5i)43-s + (39.8 − 23.0i)47-s + ⋯ |
| L(s) = 1 | + (−0.923 + 1.59i)5-s + (0.761 − 0.439i)7-s + (−0.336 + 0.194i)11-s + (0.0668 − 0.115i)13-s − 0.726·17-s + 1.78i·19-s + (−0.145 − 0.0841i)23-s + (−1.20 − 2.08i)25-s + (−0.615 − 1.06i)29-s + (−1.25 − 0.723i)31-s + 1.62i·35-s − 0.884·37-s + (−0.533 + 0.923i)41-s + (−0.789 + 0.455i)43-s + (0.848 − 0.489i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 - 0.195i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0628967 + 0.638647i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0628967 + 0.638647i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (4.61 - 7.99i)T + (-12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (-5.33 + 3.07i)T + (24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (3.70 - 2.13i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-0.869 + 1.50i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + 12.3T + 289T^{2} \) |
| 19 | \( 1 - 33.9iT - 361T^{2} \) |
| 23 | \( 1 + (3.35 + 1.93i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (17.8 + 30.9i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (38.8 + 22.4i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 32.7T + 1.36e3T^{2} \) |
| 41 | \( 1 + (21.8 - 37.8i)T + (-840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (33.9 - 19.5i)T + (924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-39.8 + 23.0i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + 46.3T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-23.2 - 13.4i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-23.4 - 40.6i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-56.9 - 32.9i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 96.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 14.0T + 5.32e3T^{2} \) |
| 79 | \( 1 + (34.3 - 19.8i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-81.7 + 47.1i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 - 81.8T + 7.92e3T^{2} \) |
| 97 | \( 1 + (7.99 + 13.8i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−11.29116199915874264177419668564, −10.59391677928643263664401659527, −9.882694078524627775734752142584, −8.257909453381068001638964705870, −7.68877650316411074170431318526, −6.90145113208726485801030315470, −5.81053834660903023922061748069, −4.27300813959336314956887172240, −3.47974582198603924186204931899, −2.08024164042651892094656961505,
0.26274374882187463889834684010, 1.81178265718805716799345010877, 3.64913838908566525434758716965, 4.93654613626675237867930324325, 5.18843106405136734162138790756, 6.95810554103591861710892646005, 7.941373943955692989059854137431, 8.900864229334221158597560620151, 9.041810808925528979719721766743, 10.82365672548377558318274841892
