| L(s) = 1 | + (0.327 + 0.945i)2-s + (−0.458 + 0.888i)3-s + (−0.786 + 0.618i)4-s + (4.16 + 0.397i)5-s + (−0.989 − 0.142i)6-s + (0.0805 − 2.64i)7-s + (−0.841 − 0.540i)8-s + (−0.580 − 0.814i)9-s + (0.986 + 4.06i)10-s + (−0.191 − 0.0663i)11-s + (−0.189 − 0.981i)12-s + (−1.51 + 5.17i)13-s + (2.52 − 0.788i)14-s + (−2.26 + 3.52i)15-s + (0.235 − 0.971i)16-s + (−4.53 + 1.81i)17-s + ⋯ |
| L(s) = 1 | + (0.231 + 0.668i)2-s + (−0.264 + 0.513i)3-s + (−0.393 + 0.309i)4-s + (1.86 + 0.177i)5-s + (−0.404 − 0.0580i)6-s + (0.0304 − 0.999i)7-s + (−0.297 − 0.191i)8-s + (−0.193 − 0.271i)9-s + (0.312 + 1.28i)10-s + (−0.0577 − 0.0199i)11-s + (−0.0546 − 0.283i)12-s + (−0.421 + 1.43i)13-s + (0.674 − 0.210i)14-s + (−0.584 + 0.909i)15-s + (0.0589 − 0.242i)16-s + (−1.09 + 0.439i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.115 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.115 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.40353 + 1.57614i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.40353 + 1.57614i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.327 - 0.945i)T \) |
| 3 | \( 1 + (0.458 - 0.888i)T \) |
| 7 | \( 1 + (-0.0805 + 2.64i)T \) |
| 23 | \( 1 + (-4.79 + 0.0590i)T \) |
| good | 5 | \( 1 + (-4.16 - 0.397i)T + (4.90 + 0.946i)T^{2} \) |
| 11 | \( 1 + (0.191 + 0.0663i)T + (8.64 + 6.79i)T^{2} \) |
| 13 | \( 1 + (1.51 - 5.17i)T + (-10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (4.53 - 1.81i)T + (12.3 - 11.7i)T^{2} \) |
| 19 | \( 1 + (-3.26 - 1.30i)T + (13.7 + 13.1i)T^{2} \) |
| 29 | \( 1 + (0.707 - 4.91i)T + (-27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (-7.82 - 0.372i)T + (30.8 + 2.94i)T^{2} \) |
| 37 | \( 1 + (-0.879 + 0.626i)T + (12.1 - 34.9i)T^{2} \) |
| 41 | \( 1 + (2.62 - 1.19i)T + (26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (-3.34 - 5.20i)T + (-17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + (-6.61 - 3.81i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.18 + 5.43i)T + (-2.52 - 52.9i)T^{2} \) |
| 59 | \( 1 + (9.75 - 2.36i)T + (52.4 - 27.0i)T^{2} \) |
| 61 | \( 1 + (5.15 - 2.65i)T + (35.3 - 49.6i)T^{2} \) |
| 67 | \( 1 + (-2.09 + 10.8i)T + (-62.2 - 24.9i)T^{2} \) |
| 71 | \( 1 + (0.0174 - 0.0201i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (6.57 + 8.35i)T + (-17.2 + 70.9i)T^{2} \) |
| 79 | \( 1 + (11.2 + 11.7i)T + (-3.75 + 78.9i)T^{2} \) |
| 83 | \( 1 + (-2.10 + 4.61i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (0.851 + 17.8i)T + (-88.5 + 8.45i)T^{2} \) |
| 97 | \( 1 + (-0.146 - 0.319i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−10.20002275987643518175222047544, −9.296989870655800868987581621995, −8.943079773817086416515425759613, −7.41014134055542205028763651078, −6.60302939096078644575704072954, −6.11737633422254186102237775296, −4.96938010183692141537760458437, −4.41790060266786597967338410328, −2.97058208040942605884474541146, −1.54521121956127400412314571178,
1.04152375191357281607644659567, 2.44998082761253187135990805081, 2.70307141083735519411170467899, 4.83282920147797419995459804031, 5.48374741153471837107504663861, 6.02610783133198924828183897899, 7.05485132944321026551131966206, 8.427536209016755234714463516721, 9.157310918321573950048726752941, 9.850047583490697204394387924670
