| L(s) = 1 | + (−0.388 − 0.309i)2-s + (0.252 − 1.71i)3-s + (−0.390 − 1.70i)4-s + (0.146 + 1.95i)5-s + (−0.629 + 0.587i)6-s + (−2.18 + 1.49i)7-s + (−0.809 + 1.68i)8-s + (−2.87 − 0.865i)9-s + (0.549 − 0.806i)10-s + (−1.46 + 0.575i)11-s + (−3.02 + 0.236i)12-s + (−5.93 + 2.33i)13-s + (1.31 + 0.0975i)14-s + (3.39 + 0.243i)15-s + (−2.32 + 1.11i)16-s + (3.85 − 1.18i)17-s + ⋯ |
| L(s) = 1 | + (−0.274 − 0.219i)2-s + (0.145 − 0.989i)3-s + (−0.195 − 0.854i)4-s + (0.0656 + 0.876i)5-s + (−0.256 + 0.239i)6-s + (−0.825 + 0.563i)7-s + (−0.286 + 0.594i)8-s + (−0.957 − 0.288i)9-s + (0.173 − 0.255i)10-s + (−0.442 + 0.173i)11-s + (−0.873 + 0.0683i)12-s + (−1.64 + 0.646i)13-s + (0.350 + 0.0260i)14-s + (0.876 + 0.0628i)15-s + (−0.580 + 0.279i)16-s + (0.935 − 0.288i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.607 - 0.794i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.607 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0252367 + 0.0510326i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0252367 + 0.0510326i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.252 + 1.71i)T \) |
| 7 | \( 1 + (2.18 - 1.49i)T \) |
| good | 2 | \( 1 + (0.388 + 0.309i)T + (0.445 + 1.94i)T^{2} \) |
| 5 | \( 1 + (-0.146 - 1.95i)T + (-4.94 + 0.745i)T^{2} \) |
| 11 | \( 1 + (1.46 - 0.575i)T + (8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (5.93 - 2.33i)T + (9.52 - 8.84i)T^{2} \) |
| 17 | \( 1 + (-3.85 + 1.18i)T + (14.0 - 9.57i)T^{2} \) |
| 19 | \( 1 + (3.00 + 1.73i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (4.83 + 5.20i)T + (-1.71 + 22.9i)T^{2} \) |
| 29 | \( 1 + (0.915 + 2.96i)T + (-23.9 + 16.3i)T^{2} \) |
| 31 | \( 1 - 2.10iT - 31T^{2} \) |
| 37 | \( 1 + (-5.36 - 4.98i)T + (2.76 + 36.8i)T^{2} \) |
| 41 | \( 1 + (-2.05 + 1.40i)T + (14.9 - 38.1i)T^{2} \) |
| 43 | \( 1 + (4.55 + 3.10i)T + (15.7 + 40.0i)T^{2} \) |
| 47 | \( 1 + (2.93 - 3.68i)T + (-10.4 - 45.8i)T^{2} \) |
| 53 | \( 1 + (4.89 + 5.27i)T + (-3.96 + 52.8i)T^{2} \) |
| 59 | \( 1 + (-1.16 + 0.561i)T + (36.7 - 46.1i)T^{2} \) |
| 61 | \( 1 + (3.49 + 0.797i)T + (54.9 + 26.4i)T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 + (-8.98 + 2.04i)T + (63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (3.59 + 1.41i)T + (53.5 + 49.6i)T^{2} \) |
| 79 | \( 1 + 12.4T + 79T^{2} \) |
| 83 | \( 1 + (0.693 - 1.76i)T + (-60.8 - 56.4i)T^{2} \) |
| 89 | \( 1 + (-12.3 + 1.85i)T + (85.0 - 26.2i)T^{2} \) |
| 97 | \( 1 + (-2.27 + 1.31i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.32958513562953401948106503981, −9.788548692835665735730189288969, −8.839843773783449554510107174083, −7.68562476995885541199821995286, −6.67693705059668622173903912452, −6.10957742432177638947959300264, −4.88910496538794565567867459769, −2.83561816409914661814348421610, −2.16940552047255744329238953722, −0.03525408656550642778161961261,
2.88905450354606677747470123116, 3.86464426345751765099356187034, 4.84817419354109036021781514226, 5.91147463952753813732874151146, 7.50848641306238606700616587305, 8.044851478124288727157522901110, 9.142268047016288639341336457106, 9.776708158374339983744330816926, 10.41901681972866282419441480715, 11.81499538855908279838403312259
