| L(s) = 1 | + (0.843 − 1.13i)2-s + (0.935 + 1.45i)3-s + (−0.575 − 1.91i)4-s + (−1.06 + 2.92i)5-s + (2.44 + 0.169i)6-s + (−0.984 − 0.173i)7-s + (−2.65 − 0.962i)8-s + (−1.25 + 2.72i)9-s + (2.41 + 3.67i)10-s + (0.301 − 0.109i)11-s + (2.25 − 2.63i)12-s + (−4.68 + 3.93i)13-s + (−1.02 + 0.971i)14-s + (−5.25 + 1.18i)15-s + (−3.33 + 2.20i)16-s + (−4.28 + 2.47i)17-s + ⋯ |
| L(s) = 1 | + (0.596 − 0.802i)2-s + (0.539 + 0.841i)3-s + (−0.287 − 0.957i)4-s + (−0.475 + 1.30i)5-s + (0.997 + 0.0690i)6-s + (−0.372 − 0.0656i)7-s + (−0.940 − 0.340i)8-s + (−0.417 + 0.908i)9-s + (0.764 + 1.16i)10-s + (0.0909 − 0.0331i)11-s + (0.650 − 0.759i)12-s + (−1.29 + 1.09i)13-s + (−0.274 + 0.259i)14-s + (−1.35 + 0.304i)15-s + (−0.834 + 0.551i)16-s + (−1.03 + 0.599i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0247 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0247 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.02967 + 1.05544i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.02967 + 1.05544i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.843 + 1.13i)T \) |
| 3 | \( 1 + (-0.935 - 1.45i)T \) |
| 7 | \( 1 + (0.984 + 0.173i)T \) |
| good | 5 | \( 1 + (1.06 - 2.92i)T + (-3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 + (-0.301 + 0.109i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (4.68 - 3.93i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (4.28 - 2.47i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-6.34 - 3.66i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.18 + 6.73i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-4.53 + 5.40i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.171 + 0.0303i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-1.39 - 2.41i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.72 - 3.24i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-2.01 - 5.52i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (0.954 - 5.41i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 1.61iT - 53T^{2} \) |
| 59 | \( 1 + (-4.65 - 1.69i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.585 + 3.32i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-9.54 - 11.3i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (1.41 + 2.45i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.23 - 3.87i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-8.71 + 10.3i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (5.07 + 4.25i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (0.0619 + 0.0357i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.95 + 1.07i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.49217918340907022081294019328, −9.937608743699359057629572078590, −9.285690122935379194967224747641, −8.072958375819830244246187272634, −6.93733103755501469602599021251, −6.09676570475484770987765269649, −4.65618442467365028252415930066, −4.05613880993063287662559139754, −2.99033643498655572774511513561, −2.33295065062477802214170049698,
0.55365044331480058343063013388, 2.60530219211375449251696086342, 3.64348163225401475998138233892, 4.96502065806232244522923978360, 5.46978468629895145634811109527, 6.90868676680578206161660841986, 7.43280646253233086364193403796, 8.209045460411192884590526630305, 9.055425823703848934729071300892, 9.578954210165319010104023930934
