L(s) = 1 | + (0.173 + 0.984i)2-s + (0.854 − 0.717i)3-s + (−0.939 + 0.342i)4-s + (−1.23 − 0.449i)5-s + (0.854 + 0.717i)6-s + (−2.13 + 3.70i)7-s + (−0.5 − 0.866i)8-s + (−0.304 + 1.72i)9-s + (0.227 − 1.29i)10-s + (0.5 + 0.866i)11-s + (−0.557 + 0.966i)12-s + (0.986 + 0.827i)13-s + (−4.01 − 1.46i)14-s + (−1.37 + 0.501i)15-s + (0.766 − 0.642i)16-s + (0.723 + 4.10i)17-s + ⋯ |
L(s) = 1 | + (0.122 + 0.696i)2-s + (0.493 − 0.414i)3-s + (−0.469 + 0.171i)4-s + (−0.551 − 0.200i)5-s + (0.349 + 0.292i)6-s + (−0.807 + 1.39i)7-s + (−0.176 − 0.306i)8-s + (−0.101 + 0.575i)9-s + (0.0720 − 0.408i)10-s + (0.150 + 0.261i)11-s + (−0.161 + 0.278i)12-s + (0.273 + 0.229i)13-s + (−1.07 − 0.390i)14-s + (−0.355 + 0.129i)15-s + (0.191 − 0.160i)16-s + (0.175 + 0.994i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.616 - 0.787i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.616 - 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.492728 + 1.01229i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.492728 + 1.01229i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.173 - 0.984i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 19 | \( 1 + (4.32 - 0.545i)T \) |
good | 3 | \( 1 + (-0.854 + 0.717i)T + (0.520 - 2.95i)T^{2} \) |
| 5 | \( 1 + (1.23 + 0.449i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (2.13 - 3.70i)T + (-3.5 - 6.06i)T^{2} \) |
| 13 | \( 1 + (-0.986 - 0.827i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.723 - 4.10i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (-4.86 + 1.76i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (0.771 - 4.37i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (0.707 - 1.22i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 9.94T + 37T^{2} \) |
| 41 | \( 1 + (-1.09 + 0.919i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (7.34 + 2.67i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.24 + 7.06i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (6.61 - 2.40i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-0.314 - 1.78i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-5.07 + 1.84i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.669 + 3.79i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-4.55 - 1.65i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-8.09 + 6.78i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-5.53 + 4.64i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (5.90 - 10.2i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (1.59 + 1.34i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (1.63 + 9.28i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−11.76871065607407164425629653790, −10.57394537048435077910997883661, −9.301140702466397830128854098650, −8.582625828219102202030811994161, −8.016161056482066540131136173137, −6.80585900619408312205775000860, −6.00021037708405898568284368035, −4.87154357052639810334478934919, −3.53075903140029468260791009912, −2.20924289344803234149381147195,
0.66189199612107455702271442521, 2.96794137774737825790401086590, 3.69906342745920944400302911071, 4.48880677564078540229390144740, 6.18913439764443669234461272236, 7.21379676904942307254237963359, 8.247994843686182935832774000840, 9.468048184583303844112435621320, 9.842721806138236495075099091796, 11.00602579164428829876732999081