| L(s) = 1 | + (−1.37 + 0.349i)2-s + (−0.514 − 0.857i)3-s + (1.75 − 0.957i)4-s + (−0.488 + 1.36i)5-s + (1.00 + 0.995i)6-s + (−3.55 − 1.07i)7-s + (−2.07 + 1.92i)8-s + (−0.471 + 0.881i)9-s + (0.192 − 2.04i)10-s + (−2.34 + 1.74i)11-s + (−1.72 − 1.01i)12-s + (0.904 − 1.91i)13-s + (5.24 + 0.235i)14-s + (1.42 − 0.282i)15-s + (2.16 − 3.36i)16-s + (2.99 + 0.594i)17-s + ⋯ |
| L(s) = 1 | + (−0.969 + 0.247i)2-s + (−0.296 − 0.495i)3-s + (0.877 − 0.478i)4-s + (−0.218 + 0.610i)5-s + (0.409 + 0.406i)6-s + (−1.34 − 0.407i)7-s + (−0.732 + 0.680i)8-s + (−0.157 + 0.293i)9-s + (0.0608 − 0.645i)10-s + (−0.708 + 0.525i)11-s + (−0.497 − 0.292i)12-s + (0.250 − 0.530i)13-s + (1.40 + 0.0630i)14-s + (0.366 − 0.0729i)15-s + (0.541 − 0.840i)16-s + (0.725 + 0.144i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.138i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 + 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.635481 - 0.0440973i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.635481 - 0.0440973i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.37 - 0.349i)T \) |
| 3 | \( 1 + (0.514 + 0.857i)T \) |
| good | 5 | \( 1 + (0.488 - 1.36i)T + (-3.86 - 3.17i)T^{2} \) |
| 7 | \( 1 + (3.55 + 1.07i)T + (5.82 + 3.88i)T^{2} \) |
| 11 | \( 1 + (2.34 - 1.74i)T + (3.19 - 10.5i)T^{2} \) |
| 13 | \( 1 + (-0.904 + 1.91i)T + (-8.24 - 10.0i)T^{2} \) |
| 17 | \( 1 + (-2.99 - 0.594i)T + (15.7 + 6.50i)T^{2} \) |
| 19 | \( 1 + (-1.49 - 0.0736i)T + (18.9 + 1.86i)T^{2} \) |
| 23 | \( 1 + (-0.0153 + 0.155i)T + (-22.5 - 4.48i)T^{2} \) |
| 29 | \( 1 + (-6.33 + 0.940i)T + (27.7 - 8.41i)T^{2} \) |
| 31 | \( 1 + (-3.16 + 1.31i)T + (21.9 - 21.9i)T^{2} \) |
| 37 | \( 1 + (2.57 + 2.33i)T + (3.62 + 36.8i)T^{2} \) |
| 41 | \( 1 + (-6.81 + 5.59i)T + (7.99 - 40.2i)T^{2} \) |
| 43 | \( 1 + (6.61 + 3.96i)T + (20.2 + 37.9i)T^{2} \) |
| 47 | \( 1 + (1.88 - 1.25i)T + (17.9 - 43.4i)T^{2} \) |
| 53 | \( 1 + (-4.62 - 0.685i)T + (50.7 + 15.3i)T^{2} \) |
| 59 | \( 1 + (-5.94 - 12.5i)T + (-37.4 + 45.6i)T^{2} \) |
| 61 | \( 1 + (-1.45 + 5.80i)T + (-53.7 - 28.7i)T^{2} \) |
| 67 | \( 1 + (-10.9 - 2.73i)T + (59.0 + 31.5i)T^{2} \) |
| 71 | \( 1 + (-3.42 + 1.83i)T + (39.4 - 59.0i)T^{2} \) |
| 73 | \( 1 + (-4.45 + 1.35i)T + (60.6 - 40.5i)T^{2} \) |
| 79 | \( 1 + (-6.70 + 10.0i)T + (-30.2 - 72.9i)T^{2} \) |
| 83 | \( 1 + (4.73 - 4.29i)T + (8.13 - 82.6i)T^{2} \) |
| 89 | \( 1 + (0.0378 + 0.384i)T + (-87.2 + 17.3i)T^{2} \) |
| 97 | \( 1 + (-6.09 - 14.7i)T + (-68.5 + 68.5i)T^{2} \) |
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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
−10.28978343791706508220204289891, −9.637390714135991332727161209613, −8.474529326417532673364396146938, −7.56358319694773686557432678059, −6.98549541167798299036189233427, −6.25169343332249546351810452977, −5.31148906574690600381287190609, −3.44009751496554834583274195362, −2.51138384113061582418524739016, −0.72010732393510521649195020273,
0.78688644256327558876033059741, 2.74628778411821167954904958107, 3.55851008135073330972344413987, 5.00345189694431082888656474125, 6.13224827585769531432691539414, 6.81903286518099872681730333328, 8.146139929052243071462319817593, 8.695256456596849710947994901432, 9.694133527985686969861639582226, 10.02180534584476062042487066072
