| L(s) = 1 | + (−4.03 − 2.33i)2-s + (−1.78 + 4.87i)3-s + (6.86 + 11.8i)4-s + (−8.94 − 2.39i)5-s + (18.5 − 15.5i)6-s + (−27.6 + 7.39i)7-s − 26.6i·8-s + (−20.6 − 17.4i)9-s + (30.5 + 30.5i)10-s + (−42.0 + 11.2i)11-s + (−70.2 + 12.2i)12-s + (22.2 + 38.4i)13-s + (128. + 34.4i)14-s + (27.6 − 39.3i)15-s + (−7.28 + 12.6i)16-s + (28.9 − 63.8i)17-s + ⋯ |
| L(s) = 1 | + (−1.42 − 0.823i)2-s + (−0.343 + 0.939i)3-s + (0.857 + 1.48i)4-s + (−0.799 − 0.214i)5-s + (1.26 − 1.05i)6-s + (−1.49 + 0.399i)7-s − 1.17i·8-s + (−0.763 − 0.645i)9-s + (0.965 + 0.965i)10-s + (−1.15 + 0.308i)11-s + (−1.69 + 0.294i)12-s + (0.474 + 0.821i)13-s + (2.45 + 0.658i)14-s + (0.476 − 0.677i)15-s + (−0.113 + 0.197i)16-s + (0.413 − 0.910i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.396 + 0.918i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.396 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.158964 - 0.104530i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.158964 - 0.104530i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.78 - 4.87i)T \) |
| 17 | \( 1 + (-28.9 + 63.8i)T \) |
| good | 2 | \( 1 + (4.03 + 2.33i)T + (4 + 6.92i)T^{2} \) |
| 5 | \( 1 + (8.94 + 2.39i)T + (108. + 62.5i)T^{2} \) |
| 7 | \( 1 + (27.6 - 7.39i)T + (297. - 171.5i)T^{2} \) |
| 11 | \( 1 + (42.0 - 11.2i)T + (1.15e3 - 665.5i)T^{2} \) |
| 13 | \( 1 + (-22.2 - 38.4i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 19 | \( 1 - 159. iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (-35.9 + 134. i)T + (-1.05e4 - 6.08e3i)T^{2} \) |
| 29 | \( 1 + (-34.3 - 128. i)T + (-2.11e4 + 1.21e4i)T^{2} \) |
| 31 | \( 1 + (165. + 44.2i)T + (2.57e4 + 1.48e4i)T^{2} \) |
| 37 | \( 1 + (-287. + 287. i)T - 5.06e4iT^{2} \) |
| 41 | \( 1 + (-57.2 + 213. i)T + (-5.96e4 - 3.44e4i)T^{2} \) |
| 43 | \( 1 + (-117. - 68.0i)T + (3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-79.0 + 136. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 - 459. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (198. - 114. i)T + (1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-93.2 + 24.9i)T + (1.96e5 - 1.13e5i)T^{2} \) |
| 67 | \( 1 + (-90.4 - 156. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + (614. - 614. i)T - 3.57e5iT^{2} \) |
| 73 | \( 1 + (-166. + 166. i)T - 3.89e5iT^{2} \) |
| 79 | \( 1 + (-49.8 + 13.3i)T + (4.26e5 - 2.46e5i)T^{2} \) |
| 83 | \( 1 + (563. + 325. i)T + (2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + 203.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (58.2 + 217. i)T + (-7.90e5 + 4.56e5i)T^{2} \) |
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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
−12.10822385243831052027117783580, −10.98188287594042935135250123543, −10.19183630682478612834372300856, −9.442983597309369104080079575089, −8.626343927676509806398699164580, −7.42175610427593636339803390548, −5.79292649692948441896570725607, −3.94356964080220302870989051407, −2.75975884243094384215830520462, −0.25877028201605910757778739410,
0.68765306867582716019540779973, 3.11018052870099998590621135324, 5.72247061919949788639233253987, 6.66061034120798830457384649196, 7.55982061679133201813479516947, 8.156552127338205318503567844741, 9.457764725769451330756220887994, 10.56222071201467135948309943301, 11.33224378368071120204735009083, 12.98452534511349969874850190897
