| L(s) = 1 | + (−0.536 − 1.64i)3-s + (−1.32 − 1.57i)5-s + (−1.34 − 3.70i)7-s + (−2.42 + 1.76i)9-s + (1.12 + 0.948i)11-s + (0.716 + 4.06i)13-s + (−1.88 + 3.02i)15-s + (−4.21 + 2.43i)17-s + (0.0640 + 0.0369i)19-s + (−5.37 + 4.20i)21-s + (−7.06 − 2.56i)23-s + (0.134 − 0.760i)25-s + (4.20 + 3.04i)27-s + (4.84 + 0.855i)29-s + (2.86 − 7.87i)31-s + ⋯ |
| L(s) = 1 | + (−0.309 − 0.950i)3-s + (−0.591 − 0.704i)5-s + (−0.509 − 1.39i)7-s + (−0.808 + 0.588i)9-s + (0.340 + 0.285i)11-s + (0.198 + 1.12i)13-s + (−0.486 + 0.780i)15-s + (−1.02 + 0.590i)17-s + (0.0146 + 0.00848i)19-s + (−1.17 + 0.917i)21-s + (−1.47 − 0.535i)23-s + (0.0268 − 0.152i)25-s + (0.810 + 0.586i)27-s + (0.900 + 0.158i)29-s + (0.514 − 1.41i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.160i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 - 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0454336 + 0.563459i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0454336 + 0.563459i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.536 + 1.64i)T \) |
| good | 5 | \( 1 + (1.32 + 1.57i)T + (-0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (1.34 + 3.70i)T + (-5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (-1.12 - 0.948i)T + (1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.716 - 4.06i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (4.21 - 2.43i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.0640 - 0.0369i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (7.06 + 2.56i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-4.84 - 0.855i)T + (27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-2.86 + 7.87i)T + (-23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (2.58 + 4.47i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (8.47 - 1.49i)T + (38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-0.636 + 0.758i)T + (-7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (6.18 - 2.25i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + 7.92iT - 53T^{2} \) |
| 59 | \( 1 + (-0.418 + 0.351i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-11.8 + 4.29i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (11.0 - 1.94i)T + (62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (3.59 + 6.23i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.52 + 7.83i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.35 - 0.414i)T + (74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (0.677 - 3.84i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (-14.2 - 8.23i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-10.1 - 8.50i)T + (16.8 + 95.5i)T^{2} \) |
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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
−10.83203718312578188247187488530, −9.842414097972368502002777341947, −8.600226358163919479664249622361, −7.899127650432330329404129000543, −6.79676784564145691917119284327, −6.35254175456163346440340658071, −4.61236365180392942364923116718, −3.90067852815681468755274847437, −1.89311854384948623699582908632, −0.36096741991972578308763146303,
2.79071056624571058350184477085, 3.54383452053187639275126670209, 4.95388971394729229510980391233, 5.91087207954881583938517693827, 6.74540873300875450840327769001, 8.265122185213636492341972715655, 8.915878760592299982496014660639, 9.962540240005852843636537394147, 10.65357312300033534463575238011, 11.81647393481053163472050812690
