| L(s) = 1 | + (0.275 + 0.263i)2-s + (−0.981 + 0.189i)3-s + (−0.0882 − 1.85i)4-s + (−3.33 + 2.62i)5-s + (−0.320 − 0.206i)6-s + (2.63 + 0.234i)7-s + (0.962 − 1.11i)8-s + (0.928 − 0.371i)9-s + (−1.60 − 0.153i)10-s + (−0.117 + 0.112i)11-s + (0.437 + 1.80i)12-s + (−2.17 − 4.75i)13-s + (0.665 + 0.757i)14-s + (2.77 − 3.20i)15-s + (−3.13 + 0.299i)16-s + (−1.07 + 0.552i)17-s + ⋯ |
| L(s) = 1 | + (0.195 + 0.186i)2-s + (−0.566 + 0.109i)3-s + (−0.0441 − 0.926i)4-s + (−1.49 + 1.17i)5-s + (−0.130 − 0.0841i)6-s + (0.996 + 0.0886i)7-s + (0.340 − 0.392i)8-s + (0.309 − 0.123i)9-s + (−0.508 − 0.0485i)10-s + (−0.0355 + 0.0338i)11-s + (0.126 + 0.520i)12-s + (−0.602 − 1.32i)13-s + (0.177 + 0.202i)14-s + (0.716 − 0.827i)15-s + (−0.783 + 0.0748i)16-s + (−0.260 + 0.134i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.477 + 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.477 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.252304 - 0.424142i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.252304 - 0.424142i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.981 - 0.189i)T \) |
| 7 | \( 1 + (-2.63 - 0.234i)T \) |
| 23 | \( 1 + (-1.51 + 4.54i)T \) |
| good | 2 | \( 1 + (-0.275 - 0.263i)T + (0.0951 + 1.99i)T^{2} \) |
| 5 | \( 1 + (3.33 - 2.62i)T + (1.17 - 4.85i)T^{2} \) |
| 11 | \( 1 + (0.117 - 0.112i)T + (0.523 - 10.9i)T^{2} \) |
| 13 | \( 1 + (2.17 + 4.75i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (1.07 - 0.552i)T + (9.86 - 13.8i)T^{2} \) |
| 19 | \( 1 + (7.46 + 3.84i)T + (11.0 + 15.4i)T^{2} \) |
| 29 | \( 1 + (4.84 + 3.11i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (-0.678 - 1.96i)T + (-24.3 + 19.1i)T^{2} \) |
| 37 | \( 1 + (1.50 - 0.602i)T + (26.7 - 25.5i)T^{2} \) |
| 41 | \( 1 + (0.616 + 4.28i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (-4.14 - 4.78i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 + (-0.923 - 1.60i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.866 - 1.21i)T + (-17.3 + 50.0i)T^{2} \) |
| 59 | \( 1 + (-2.04 - 0.195i)T + (57.9 + 11.1i)T^{2} \) |
| 61 | \( 1 + (-0.139 - 0.0269i)T + (56.6 + 22.6i)T^{2} \) |
| 67 | \( 1 + (2.54 - 10.5i)T + (-59.5 - 30.7i)T^{2} \) |
| 71 | \( 1 + (11.4 - 3.35i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (0.465 + 9.76i)T + (-72.6 + 6.93i)T^{2} \) |
| 79 | \( 1 + (-7.11 + 9.99i)T + (-25.8 - 74.6i)T^{2} \) |
| 83 | \( 1 + (-1.72 + 11.9i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (2.94 - 8.50i)T + (-69.9 - 55.0i)T^{2} \) |
| 97 | \( 1 + (-0.197 - 1.37i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−10.68131182191964468377196205847, −10.40690687032849573159227076037, −8.755392697121400159707823174711, −7.73957297278254388885642922837, −6.98134724412713183798984325833, −6.05516669716672534211552524947, −4.83906689025508450949469310277, −4.16634059974850777829174728309, −2.52316957165185772419322830727, −0.29823262835910006042850321538,
1.80580367061783567588852148694, 3.92364036693625064053574628736, 4.36263837852549813400180931408, 5.22648619266919987713575114336, 7.00218493200020470780966096441, 7.72911746867525770028591382161, 8.445392222455188856068985148028, 9.174808775312840803902415405561, 10.93878192257329824262944358035, 11.47120513232377570732380618258
