| L(s) = 1 | + (−1.85 − 1.19i)2-s + (0.989 − 0.142i)3-s + (1.19 + 2.61i)4-s + (−1.10 − 0.324i)5-s + (−2.00 − 0.916i)6-s + (1.07 − 2.41i)7-s + (0.274 − 1.90i)8-s + (0.959 − 0.281i)9-s + (1.66 + 1.92i)10-s + (−0.0983 − 0.152i)11-s + (1.55 + 2.41i)12-s + (3.11 − 2.70i)13-s + (−4.88 + 3.19i)14-s + (−1.14 − 0.164i)15-s + (0.978 − 1.12i)16-s + (−2.24 + 4.91i)17-s + ⋯ |
| L(s) = 1 | + (−1.31 − 0.843i)2-s + (0.571 − 0.0821i)3-s + (0.596 + 1.30i)4-s + (−0.494 − 0.145i)5-s + (−0.819 − 0.374i)6-s + (0.407 − 0.913i)7-s + (0.0969 − 0.674i)8-s + (0.319 − 0.0939i)9-s + (0.527 + 0.608i)10-s + (−0.0296 − 0.0461i)11-s + (0.448 + 0.697i)12-s + (0.864 − 0.749i)13-s + (−1.30 + 0.854i)14-s + (−0.294 − 0.0423i)15-s + (0.244 − 0.282i)16-s + (−0.544 + 1.19i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.713 + 0.701i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.713 + 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.287182 - 0.701758i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.287182 - 0.701758i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.989 + 0.142i)T \) |
| 7 | \( 1 + (-1.07 + 2.41i)T \) |
| 23 | \( 1 + (-0.108 + 4.79i)T \) |
| good | 2 | \( 1 + (1.85 + 1.19i)T + (0.830 + 1.81i)T^{2} \) |
| 5 | \( 1 + (1.10 + 0.324i)T + (4.20 + 2.70i)T^{2} \) |
| 11 | \( 1 + (0.0983 + 0.152i)T + (-4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (-3.11 + 2.70i)T + (1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (2.24 - 4.91i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (0.214 + 0.469i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (-3.47 + 7.60i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (-3.48 - 0.500i)T + (29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (0.139 + 0.473i)T + (-31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (-2.65 + 9.05i)T + (-34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (9.24 - 1.32i)T + (41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 - 3.91iT - 47T^{2} \) |
| 53 | \( 1 + (5.66 + 4.91i)T + (7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-3.29 + 2.85i)T + (8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (0.757 - 5.26i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (1.30 - 2.03i)T + (-27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (7.91 + 5.08i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (5.88 - 2.68i)T + (47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (-2.99 + 2.59i)T + (11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-8.62 + 2.53i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (-2.58 - 18.0i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (-6.91 - 2.02i)T + (81.6 + 52.4i)T^{2} \) |
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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
−10.49054317446101399285677605401, −9.964821122125420124572120537837, −8.631317956977326875806934773478, −8.284214098607259496061370683434, −7.58201118767935316270728680982, −6.28499708725321515276971053841, −4.42934290770682009573117668957, −3.44270036293876870259369816346, −2.05942472203700997633768773745, −0.70085248838139110288034668534,
1.62580134628004418938629082819, 3.27184657091619915096361999364, 4.79759409587526073958312777984, 6.12255590444873764968667764317, 7.06092448886667563358975583068, 7.86571805252360419721510001863, 8.688174922874614159985866309282, 9.155820391072733806393879240676, 10.01950731713344025574765723709, 11.22498350022909008704418385646
