| L(s) = 1 | + (0.770 − 1.18i)2-s + (0.759 + 1.55i)3-s + (−0.814 − 1.82i)4-s + (0.857 − 2.35i)5-s + (2.43 + 0.297i)6-s + (0.442 − 2.51i)7-s + (−2.79 − 0.440i)8-s + (−1.84 + 2.36i)9-s + (−2.13 − 2.83i)10-s + (1.32 + 3.65i)11-s + (2.22 − 2.65i)12-s + (1.81 + 2.15i)13-s + (−2.63 − 2.45i)14-s + (4.31 − 0.454i)15-s + (−2.67 + 2.97i)16-s + (−3.79 − 6.56i)17-s + ⋯ |
| L(s) = 1 | + (0.544 − 0.838i)2-s + (0.438 + 0.898i)3-s + (−0.407 − 0.913i)4-s + (0.383 − 1.05i)5-s + (0.992 + 0.121i)6-s + (0.167 − 0.949i)7-s + (−0.987 − 0.155i)8-s + (−0.615 + 0.788i)9-s + (−0.674 − 0.895i)10-s + (0.400 + 1.10i)11-s + (0.642 − 0.766i)12-s + (0.502 + 0.598i)13-s + (−0.705 − 0.657i)14-s + (1.11 − 0.117i)15-s + (−0.668 + 0.743i)16-s + (−0.919 − 1.59i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.434 + 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.434 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.50615 - 0.945974i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.50615 - 0.945974i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.770 + 1.18i)T \) |
| 3 | \( 1 + (-0.759 - 1.55i)T \) |
| good | 5 | \( 1 + (-0.857 + 2.35i)T + (-3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (-0.442 + 2.51i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (-1.32 - 3.65i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-1.81 - 2.15i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (3.79 + 6.56i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.25 - 2.45i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.488 - 2.77i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (1.68 - 2.00i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.605 - 3.43i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (6.91 - 3.99i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (7.68 - 6.44i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-1.38 - 3.81i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.48 + 8.41i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 11.8iT - 53T^{2} \) |
| 59 | \( 1 + (-0.584 + 1.60i)T + (-45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.343 - 0.0605i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (1.45 + 1.72i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (2.53 + 4.38i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.34 + 2.33i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.652 + 0.547i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (5.59 - 6.67i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-3.02 + 5.24i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.71 + 1.71i)T + (74.3 - 62.3i)T^{2} \) |
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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
−12.01963743021584791882518179994, −11.25820765449805735380285886472, −10.05527643471324077119742838639, −9.496886582283322344946504457184, −8.689229312197024273190313843217, −6.99964197073269544830784470955, −5.13589018660880187364349879349, −4.64403289662309274696599756028, −3.49169011459425649638648626448, −1.64153181422720337118966411681,
2.51927105723613726049452910697, 3.59285878324914920563080753479, 5.76267229652758934258235506440, 6.22891690922140281315356092439, 7.26092347150819100551604768304, 8.468477364982684405036898559647, 8.951182672066965137819249280399, 10.78152895902363142605115935695, 11.78952700291708665167084915382, 12.76193402246946386013878168653
