| L(s) = 1 | + (1.59 + 0.678i)3-s + (1.47 − 1.23i)5-s + (−0.495 + 0.180i)7-s + (2.07 + 2.16i)9-s + (−1.04 − 0.877i)11-s + (−0.231 − 1.31i)13-s + (3.18 − 0.970i)15-s + (−1.45 − 2.51i)17-s + (−4.12 + 7.14i)19-s + (−0.912 − 0.0489i)21-s + (4.64 + 1.68i)23-s + (−0.226 + 1.28i)25-s + (1.84 + 4.85i)27-s + (1.56 − 8.84i)29-s + (−7.35 − 2.67i)31-s + ⋯ |
| L(s) = 1 | + (0.920 + 0.391i)3-s + (0.658 − 0.552i)5-s + (−0.187 + 0.0682i)7-s + (0.692 + 0.720i)9-s + (−0.315 − 0.264i)11-s + (−0.0642 − 0.364i)13-s + (0.822 − 0.250i)15-s + (−0.352 − 0.609i)17-s + (−0.946 + 1.63i)19-s + (−0.199 − 0.0106i)21-s + (0.968 + 0.352i)23-s + (−0.0452 + 0.256i)25-s + (0.355 + 0.934i)27-s + (0.289 − 1.64i)29-s + (−1.32 − 0.481i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.129i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.129i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.66768 + 0.108765i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.66768 + 0.108765i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.59 - 0.678i)T \) |
| good | 5 | \( 1 + (-1.47 + 1.23i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (0.495 - 0.180i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (1.04 + 0.877i)T + (1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.231 + 1.31i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (1.45 + 2.51i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (4.12 - 7.14i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.64 - 1.68i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-1.56 + 8.84i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (7.35 + 2.67i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (-0.567 - 0.982i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.675 + 3.82i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (1.69 + 1.42i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (6.47 - 2.35i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + 8.26T + 53T^{2} \) |
| 59 | \( 1 + (-1.66 + 1.39i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-7.98 + 2.90i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (0.319 + 1.80i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-3.51 - 6.08i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.42 - 4.19i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.99 + 11.3i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-2.49 + 14.1i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (7.50 - 12.9i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-14.2 - 11.9i)T + (16.8 + 95.5i)T^{2} \) |
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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
−12.75193344158231759367497521942, −11.23429673298961837259725500023, −10.10494522706087022778378328037, −9.431839291617585626780794909424, −8.493045394700433921808003274077, −7.57179157643303673696449177167, −6.01518724854222145459921785964, −4.86065889017900067038949664733, −3.48987839982499849822802240401, −2.01946501410179897552003953713,
2.02713753019896064978472295823, 3.18168243083638922621623144342, 4.77320392060555898554880383653, 6.54478126673060795166743810268, 7.03829917120920588301546329278, 8.487656245273887373320588956474, 9.226296295950686943348120412616, 10.29202307536480720750292133261, 11.18232307179382561186203719790, 12.83688273988482876435342527795
