| L(s) = 1 | + (−0.654 − 0.755i)2-s + (−0.841 − 0.540i)3-s + (−0.142 + 0.989i)4-s + (0.856 − 1.87i)5-s + (0.142 + 0.989i)6-s + (−0.959 − 0.281i)7-s + (0.841 − 0.540i)8-s + (0.415 + 0.909i)9-s + (−1.97 + 0.580i)10-s + (−1.11 + 1.28i)11-s + (0.654 − 0.755i)12-s + (5.23 − 1.53i)13-s + (0.415 + 0.909i)14-s + (−1.73 + 1.11i)15-s + (−0.959 − 0.281i)16-s + (−0.173 − 1.20i)17-s + ⋯ |
| L(s) = 1 | + (−0.463 − 0.534i)2-s + (−0.485 − 0.312i)3-s + (−0.0711 + 0.494i)4-s + (0.382 − 0.838i)5-s + (0.0580 + 0.404i)6-s + (−0.362 − 0.106i)7-s + (0.297 − 0.191i)8-s + (0.138 + 0.303i)9-s + (−0.625 + 0.183i)10-s + (−0.336 + 0.388i)11-s + (0.189 − 0.218i)12-s + (1.45 − 0.425i)13-s + (0.111 + 0.243i)14-s + (−0.447 + 0.287i)15-s + (−0.239 − 0.0704i)16-s + (−0.0420 − 0.292i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.904 + 0.426i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.904 + 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.188389 - 0.840606i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.188389 - 0.840606i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.654 + 0.755i)T \) |
| 3 | \( 1 + (0.841 + 0.540i)T \) |
| 7 | \( 1 + (0.959 + 0.281i)T \) |
| 23 | \( 1 + (4.60 - 1.35i)T \) |
| good | 5 | \( 1 + (-0.856 + 1.87i)T + (-3.27 - 3.77i)T^{2} \) |
| 11 | \( 1 + (1.11 - 1.28i)T + (-1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-5.23 + 1.53i)T + (10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (0.173 + 1.20i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (-1.09 + 7.58i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (0.214 + 1.49i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (-1.95 + 1.25i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (1.17 + 2.56i)T + (-24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (-1.70 + 3.74i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (0.324 + 0.208i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + 8.97T + 47T^{2} \) |
| 53 | \( 1 + (1.60 + 0.470i)T + (44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (1.14 - 0.337i)T + (49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (1.91 - 1.23i)T + (25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (5.33 + 6.15i)T + (-9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (4.31 + 4.97i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (0.319 - 2.22i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (-0.365 + 0.107i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (4.47 + 9.79i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (7.68 + 4.93i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (6.31 - 13.8i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−9.615925126895403633568588200671, −8.973883887210889809667567098343, −8.146102442493960176336377685541, −7.20121064047954914236985126859, −6.21727232271198633628453038675, −5.29223187065151787869747423634, −4.33249713598006322183827559938, −3.03934119374458179424840025001, −1.69801279365533643473816628007, −0.52273496737283478435208488346,
1.53755135671199180438339533100, 3.15618678972238918775854950586, 4.17509656782327487187326332810, 5.61435228634335196961426380387, 6.17989098681295863203987429162, 6.71380728494920880769559197642, 7.981809604895151487430155391080, 8.606526648694555657195747426470, 9.744926758981134923782467062952, 10.27327656088417870795036370834
