| L(s) = 1 | + (−2.37 − 0.866i)2-s + (0.113 + 0.642i)3-s + (3.37 + 2.83i)4-s + (−1.03 + 0.866i)5-s + (0.286 − 1.62i)6-s + (−0.766 + 1.32i)7-s + (−3.05 − 5.28i)8-s + (2.41 − 0.880i)9-s + (3.20 − 1.16i)10-s + (0.592 + 1.02i)11-s + (−1.43 + 2.49i)12-s + (−0.471 + 2.67i)13-s + (2.97 − 2.49i)14-s + (−0.673 − 0.565i)15-s + (1.15 + 6.53i)16-s + (−3.64 − 1.32i)17-s + ⋯ |
| L(s) = 1 | + (−1.68 − 0.612i)2-s + (0.0654 + 0.371i)3-s + (1.68 + 1.41i)4-s + (−0.461 + 0.387i)5-s + (0.117 − 0.664i)6-s + (−0.289 + 0.501i)7-s + (−1.07 − 1.86i)8-s + (0.806 − 0.293i)9-s + (1.01 − 0.368i)10-s + (0.178 + 0.309i)11-s + (−0.415 + 0.719i)12-s + (−0.130 + 0.742i)13-s + (0.794 − 0.666i)14-s + (−0.173 − 0.145i)15-s + (0.288 + 1.63i)16-s + (−0.884 − 0.321i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.291 - 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.291 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.221677 + 0.299232i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.221677 + 0.299232i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 \) |
| good | 2 | \( 1 + (2.37 + 0.866i)T + (1.53 + 1.28i)T^{2} \) |
| 3 | \( 1 + (-0.113 - 0.642i)T + (-2.81 + 1.02i)T^{2} \) |
| 5 | \( 1 + (1.03 - 0.866i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (0.766 - 1.32i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.592 - 1.02i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.471 - 2.67i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (3.64 + 1.32i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (3.87 + 3.25i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (4.37 - 1.59i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (1.91 - 3.32i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 4.10T + 37T^{2} \) |
| 41 | \( 1 + (-1.73 - 9.83i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (6.66 - 5.59i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (0.539 - 0.196i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (2.25 + 1.89i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (3.69 + 1.34i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (3.46 + 2.90i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (3.65 - 1.32i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-5.31 + 4.45i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (-1.06 - 6.03i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (1.70 + 9.65i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (6.15 - 10.6i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.421 + 2.38i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-6.92 - 2.52i)T + (74.3 + 62.3i)T^{2} \) |
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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
−11.38069122906229939223543810062, −10.69101398906212022474455302508, −9.565338243309051361413627510855, −9.356215495288807564123499318658, −8.229117973728227037509525020244, −7.22800717386545889228407481366, −6.50378732078804085961641643607, −4.41534159207346797351127096224, −3.11443931153631829146560710792, −1.76941978793400063006969447759,
0.42801879885335729788169371781, 1.94371739001723870941018591482, 4.06437611462371434882595805748, 5.77177903540086693668818108115, 6.83692776655074485523641804345, 7.60881518120867292699187149734, 8.217067513375733962008263303182, 9.193542266077193345456477612832, 10.10365278943600106979577634291, 10.75075747965023190227886193171
