| L(s) = 1 | + (−0.939 − 1.45i)3-s + (−1.07 − 0.188i)5-s + (−0.0466 − 0.0555i)7-s + (−1.23 + 2.73i)9-s + (−0.889 − 5.04i)11-s + (−5.31 + 1.93i)13-s + (0.730 + 1.73i)15-s + (3.79 + 2.19i)17-s + (−4.96 + 2.86i)19-s + (−0.0370 + 0.119i)21-s + (−3.14 − 2.64i)23-s + (−3.58 − 1.30i)25-s + (5.13 − 0.767i)27-s + (−1.30 + 3.58i)29-s + (−2.61 + 3.11i)31-s + ⋯ |
| L(s) = 1 | + (−0.542 − 0.840i)3-s + (−0.478 − 0.0844i)5-s + (−0.0176 − 0.0209i)7-s + (−0.412 + 0.911i)9-s + (−0.268 − 1.52i)11-s + (−1.47 + 0.536i)13-s + (0.188 + 0.448i)15-s + (0.920 + 0.531i)17-s + (−1.13 + 0.657i)19-s + (−0.00808 + 0.0261i)21-s + (−0.656 − 0.550i)23-s + (−0.717 − 0.261i)25-s + (0.989 − 0.147i)27-s + (−0.242 + 0.665i)29-s + (−0.468 + 0.558i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.950 - 0.311i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.950 - 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0340307 + 0.212828i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0340307 + 0.212828i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.939 + 1.45i)T \) |
| good | 5 | \( 1 + (1.07 + 0.188i)T + (4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (0.0466 + 0.0555i)T + (-1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (0.889 + 5.04i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (5.31 - 1.93i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-3.79 - 2.19i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (4.96 - 2.86i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.14 + 2.64i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (1.30 - 3.58i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (2.61 - 3.11i)T + (-5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (1.14 - 1.98i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.494 + 1.35i)T + (-31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.128 + 0.0227i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (4.26 - 3.57i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + 10.4iT - 53T^{2} \) |
| 59 | \( 1 + (-1.69 + 9.59i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-4.96 + 4.16i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (2.28 + 6.27i)T + (-51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-3.47 + 6.02i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.77 - 4.80i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4.83 - 13.2i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (3.77 + 1.37i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (-14.4 + 8.34i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (2.74 + 15.5i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−10.81888283258336782830001592044, −9.947741488256875935040829027871, −8.429694462618049031968624877781, −7.993291070526100051506516996096, −6.87673800402805826403847399542, −5.97453979114853509370242609294, −5.01124229459051550595813837969, −3.56604201562958805664305271835, −2.02188415544648933406247451947, −0.13659284784905257028102931201,
2.50872624522362271946667089635, 4.01343349972055530047422254475, 4.82622829660452341381403049166, 5.76159707830715245753383234190, 7.16356942902432708599667362891, 7.79621630323198553072020967085, 9.305324445555330517919782194509, 9.899628096133394382892888113293, 10.59135135855683649417720904620, 11.78723698954545779033004503940
