| L(s) = 1 | + (1.34 − 0.432i)2-s + (0.0980 − 0.995i)3-s + (1.62 − 1.16i)4-s + (−1.08 + 0.577i)5-s + (−0.298 − 1.38i)6-s + (−0.0529 − 0.265i)7-s + (1.68 − 2.26i)8-s + (−0.980 − 0.195i)9-s + (−1.20 + 1.24i)10-s + (1.36 − 1.65i)11-s + (−0.998 − 1.73i)12-s + (2.20 − 4.12i)13-s + (−0.186 − 0.335i)14-s + (0.469 + 1.13i)15-s + (1.29 − 3.78i)16-s + (1.67 − 4.04i)17-s + ⋯ |
| L(s) = 1 | + (0.952 − 0.305i)2-s + (0.0565 − 0.574i)3-s + (0.813 − 0.581i)4-s + (−0.483 + 0.258i)5-s + (−0.121 − 0.564i)6-s + (−0.0199 − 0.100i)7-s + (0.596 − 0.802i)8-s + (−0.326 − 0.0650i)9-s + (−0.381 + 0.393i)10-s + (0.410 − 0.499i)11-s + (−0.288 − 0.500i)12-s + (0.611 − 1.14i)13-s + (−0.0497 − 0.0896i)14-s + (0.121 + 0.292i)15-s + (0.322 − 0.946i)16-s + (0.406 − 0.980i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.256 + 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.256 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.81755 - 1.39867i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.81755 - 1.39867i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.34 + 0.432i)T \) |
| 3 | \( 1 + (-0.0980 + 0.995i)T \) |
| good | 5 | \( 1 + (1.08 - 0.577i)T + (2.77 - 4.15i)T^{2} \) |
| 7 | \( 1 + (0.0529 + 0.265i)T + (-6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (-1.36 + 1.65i)T + (-2.14 - 10.7i)T^{2} \) |
| 13 | \( 1 + (-2.20 + 4.12i)T + (-7.22 - 10.8i)T^{2} \) |
| 17 | \( 1 + (-1.67 + 4.04i)T + (-12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (-2.18 - 7.21i)T + (-15.7 + 10.5i)T^{2} \) |
| 23 | \( 1 + (5.10 - 3.41i)T + (8.80 - 21.2i)T^{2} \) |
| 29 | \( 1 + (5.38 - 4.41i)T + (5.65 - 28.4i)T^{2} \) |
| 31 | \( 1 + (1.89 - 1.89i)T - 31iT^{2} \) |
| 37 | \( 1 + (-8.18 - 2.48i)T + (30.7 + 20.5i)T^{2} \) |
| 41 | \( 1 + (-3.17 - 4.74i)T + (-15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (-0.379 - 3.85i)T + (-42.1 + 8.38i)T^{2} \) |
| 47 | \( 1 + (6.08 + 2.51i)T + (33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (-5.46 - 4.48i)T + (10.3 + 51.9i)T^{2} \) |
| 59 | \( 1 + (-1.97 - 3.68i)T + (-32.7 + 49.0i)T^{2} \) |
| 61 | \( 1 + (6.59 + 0.649i)T + (59.8 + 11.9i)T^{2} \) |
| 67 | \( 1 + (-7.25 - 0.714i)T + (65.7 + 13.0i)T^{2} \) |
| 71 | \( 1 + (11.3 - 2.26i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (-2.93 + 14.7i)T + (-67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (-2.22 + 0.920i)T + (55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (-1.38 + 0.419i)T + (69.0 - 46.1i)T^{2} \) |
| 89 | \( 1 + (6.65 + 4.44i)T + (34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (-1.00 + 1.00i)T - 97iT^{2} \) |
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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
−11.48516482889402058115757615367, −10.52985674125689641276403889729, −9.520571452412305357142790734945, −7.943000503186634558964668116301, −7.42054223398334846674574040078, −6.08441543014166874921757512379, −5.47903932010411641511049090495, −3.80829788240054589081105841182, −3.11436239501121664930486798823, −1.35017283278507379827836475012,
2.27405149981124207591508161275, 3.97040803550556688438336814139, 4.29227903622717002864321332081, 5.65848046051044687592986443691, 6.60364019617251859002663116614, 7.69861017502927426088062917149, 8.679057781185803690818680510370, 9.672568682302739477289841684902, 10.97160485096668153770968260637, 11.60198015607164773758278265751
