| L(s) = 1 | + (0.923 + 0.362i)2-s + (−1.50 − 0.860i)3-s + (−0.745 − 0.691i)4-s + (−2.07 − 1.41i)5-s + (−1.07 − 1.33i)6-s + (0.0122 − 2.64i)7-s + (−1.29 − 2.69i)8-s + (1.52 + 2.58i)9-s + (−1.40 − 2.05i)10-s + (0.689 + 4.57i)11-s + (0.525 + 1.68i)12-s + (4.13 − 3.29i)13-s + (0.969 − 2.43i)14-s + (1.90 + 3.91i)15-s + (−0.0696 − 0.930i)16-s + (−2.67 − 0.823i)17-s + ⋯ |
| L(s) = 1 | + (0.652 + 0.256i)2-s + (−0.868 − 0.496i)3-s + (−0.372 − 0.345i)4-s + (−0.928 − 0.632i)5-s + (−0.439 − 0.546i)6-s + (0.00464 − 0.999i)7-s + (−0.458 − 0.952i)8-s + (0.506 + 0.861i)9-s + (−0.443 − 0.650i)10-s + (0.207 + 1.37i)11-s + (0.151 + 0.485i)12-s + (1.14 − 0.914i)13-s + (0.259 − 0.651i)14-s + (0.491 + 1.00i)15-s + (−0.0174 − 0.232i)16-s + (−0.647 − 0.199i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.248 + 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.248 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.503110 - 0.648420i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.503110 - 0.648420i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.50 + 0.860i)T \) |
| 7 | \( 1 + (-0.0122 + 2.64i)T \) |
| good | 2 | \( 1 + (-0.923 - 0.362i)T + (1.46 + 1.36i)T^{2} \) |
| 5 | \( 1 + (2.07 + 1.41i)T + (1.82 + 4.65i)T^{2} \) |
| 11 | \( 1 + (-0.689 - 4.57i)T + (-10.5 + 3.24i)T^{2} \) |
| 13 | \( 1 + (-4.13 + 3.29i)T + (2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (2.67 + 0.823i)T + (14.0 + 9.57i)T^{2} \) |
| 19 | \( 1 + (-0.183 + 0.105i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.18 + 3.83i)T + (-19.0 + 12.9i)T^{2} \) |
| 29 | \( 1 + (-7.26 + 1.65i)T + (26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (0.648 + 0.374i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.41 + 3.17i)T + (2.76 - 36.8i)T^{2} \) |
| 41 | \( 1 + (0.619 - 0.298i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (-4.35 - 2.09i)T + (26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (3.35 - 8.54i)T + (-34.4 - 31.9i)T^{2} \) |
| 53 | \( 1 + (-6.93 + 7.47i)T + (-3.96 - 52.8i)T^{2} \) |
| 59 | \( 1 + (-1.85 + 1.26i)T + (21.5 - 54.9i)T^{2} \) |
| 61 | \( 1 + (-3.69 - 3.98i)T + (-4.55 + 60.8i)T^{2} \) |
| 67 | \( 1 + (5.33 - 9.23i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (4.59 + 1.04i)T + (63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (-10.4 + 4.10i)T + (53.5 - 49.6i)T^{2} \) |
| 79 | \( 1 + (6.28 + 10.8i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (1.47 - 1.84i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (-4.65 - 0.701i)T + (85.0 + 26.2i)T^{2} \) |
| 97 | \( 1 - 12.7iT - 97T^{2} \) |
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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
−12.85099247882354082833134578460, −12.07613788569429399387765287466, −10.86206724232441019055714410444, −9.940107374526525612117909685342, −8.322137867191401559095605844327, −7.18948483070888516279194233166, −6.17438343311695015857178045503, −4.72361681660761490051906837075, −4.16803040086320712306005086055, −0.796762217680073612487768871418,
3.28232251601644154621174342135, 4.15572942443917674697125959944, 5.56189361768732199446653480347, 6.53775794442084725639421938304, 8.340263892673615749944298183627, 9.107714584553474884099387765842, 10.87184960806003346051113280535, 11.56404654183429021418362837186, 11.96319989335909773009692734151, 13.27383909688768235555875678280
