| L(s) = 1 | + (−0.870 + 1.11i)2-s + (−0.977 − 1.42i)3-s + (−0.484 − 1.94i)4-s + (0.0906 − 0.248i)5-s + (2.44 + 0.155i)6-s + (0.984 + 0.173i)7-s + (2.58 + 1.14i)8-s + (−1.08 + 2.79i)9-s + (0.198 + 0.317i)10-s + (−5.73 + 2.08i)11-s + (−2.30 + 2.58i)12-s + (1.01 − 0.851i)13-s + (−1.05 + 0.946i)14-s + (−0.444 + 0.113i)15-s + (−3.53 + 1.88i)16-s + (2.19 − 1.26i)17-s + ⋯ |
| L(s) = 1 | + (−0.615 + 0.788i)2-s + (−0.564 − 0.825i)3-s + (−0.242 − 0.970i)4-s + (0.0405 − 0.111i)5-s + (0.997 + 0.0634i)6-s + (0.372 + 0.0656i)7-s + (0.913 + 0.406i)8-s + (−0.363 + 0.931i)9-s + (0.0628 + 0.100i)10-s + (−1.72 + 0.628i)11-s + (−0.664 + 0.747i)12-s + (0.281 − 0.236i)13-s + (−0.280 + 0.252i)14-s + (−0.114 + 0.0293i)15-s + (−0.882 + 0.470i)16-s + (0.532 − 0.307i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0654 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0654 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.398990 + 0.426000i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.398990 + 0.426000i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.870 - 1.11i)T \) |
| 3 | \( 1 + (0.977 + 1.42i)T \) |
| 7 | \( 1 + (-0.984 - 0.173i)T \) |
| good | 5 | \( 1 + (-0.0906 + 0.248i)T + (-3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 + (5.73 - 2.08i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-1.01 + 0.851i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.19 + 1.26i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.96 + 1.70i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.696 - 3.94i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (2.88 - 3.43i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-2.67 + 0.470i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-3.66 - 6.34i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.34 - 1.59i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (1.74 + 4.78i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (1.79 - 10.2i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 7.06iT - 53T^{2} \) |
| 59 | \( 1 + (1.92 + 0.702i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (1.40 - 7.98i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-3.41 - 4.07i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-3.68 - 6.38i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.77 + 6.54i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.83 + 8.14i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-2.06 - 1.73i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-7.43 - 4.29i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (16.7 - 6.10i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.65035205897115893427141260780, −9.666071810672662482420896035386, −8.569420874885595302849556659582, −7.75524512535988502580283416796, −7.33637099138439732426421381582, −6.26470895423416919442529161654, −5.31668427170507198481712069417, −4.83687488367656424781080756825, −2.57560272871681029448151522062, −1.23364365959535954195101411071,
0.43415037988406341674392951613, 2.37063659994253070969921847009, 3.48813697588660316658952161554, 4.52127435006411754417079432864, 5.45303211921660113872459283467, 6.59312040592163685089252739219, 7.998855273548095285413411309948, 8.430307037472880776077266607825, 9.526478432299628289349214586272, 10.39824139054666667073483415917
