| L(s) = 1 | + (−0.538 + 5.16i)3-s + (−7.60 − 4.38i)5-s + (−3.39 + 18.2i)7-s + (−26.4 − 5.56i)9-s + (12.1 + 21.1i)11-s + 2.78·13-s + (26.7 − 36.9i)15-s + (−23.4 + 13.5i)17-s + (−57.2 − 33.0i)19-s + (−92.2 − 27.3i)21-s + (44.5 − 77.1i)23-s + (−23.9 − 41.5i)25-s + (42.9 − 133. i)27-s − 77.3i·29-s + (−76.0 + 43.9i)31-s + ⋯ |
| L(s) = 1 | + (−0.103 + 0.994i)3-s + (−0.679 − 0.392i)5-s + (−0.183 + 0.983i)7-s + (−0.978 − 0.205i)9-s + (0.334 + 0.578i)11-s + 0.0594·13-s + (0.460 − 0.635i)15-s + (−0.335 + 0.193i)17-s + (−0.691 − 0.398i)19-s + (−0.958 − 0.284i)21-s + (0.403 − 0.699i)23-s + (−0.191 − 0.332i)25-s + (0.306 − 0.951i)27-s − 0.495i·29-s + (−0.440 + 0.254i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.222 + 0.974i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.222 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1660137054\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1660137054\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.538 - 5.16i)T \) |
| 7 | \( 1 + (3.39 - 18.2i)T \) |
| good | 5 | \( 1 + (7.60 + 4.38i)T + (62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (-12.1 - 21.1i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 2.78T + 2.19e3T^{2} \) |
| 17 | \( 1 + (23.4 - 13.5i)T + (2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (57.2 + 33.0i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-44.5 + 77.1i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 77.3iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (76.0 - 43.9i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (107. - 186. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 197. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 251. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (-307. + 531. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-254. + 147. i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-78.1 - 135. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (215. - 373. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (414. - 239. i)T + (1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 794.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (372. + 644. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (389. + 224. i)T + (2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + 1.02e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + (578. + 334. i)T + (3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 1.67e3T + 9.12e5T^{2} \) |
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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
−10.82247997907445084468861241992, −9.936240423310823938085155725779, −8.818009779093570602397991091932, −8.514504283528078213642500070169, −6.92517639907925235625980855369, −5.74271968874040412495572722955, −4.70466596229609862687284300594, −3.83451888705056759483073419949, −2.40438129259771219781543630435, −0.06256151076751042374064897157,
1.32530001217028331336656045826, 3.04711581996488050445348538227, 4.12869392254603368590403838666, 5.74367883291896755913873232003, 6.79826141246219784689502400463, 7.45505422168855510845764897967, 8.303926052224061865181012575122, 9.459150383453243695569559251541, 10.98771714465064675481678310796, 11.14357057080246378528839935628
