| L(s) = 1 | + (−0.959 − 0.281i)2-s + (0.654 − 0.755i)3-s + (0.841 + 0.540i)4-s + (0.362 − 2.52i)5-s + (−0.841 + 0.540i)6-s + (−0.415 − 0.909i)7-s + (−0.654 − 0.755i)8-s + (−0.142 − 0.989i)9-s + (−1.05 + 2.31i)10-s + (−3.62 + 1.06i)11-s + (0.959 − 0.281i)12-s + (−1.81 + 3.98i)13-s + (0.142 + 0.989i)14-s + (−1.66 − 1.92i)15-s + (0.415 + 0.909i)16-s + (−6.25 + 4.01i)17-s + ⋯ |
| L(s) = 1 | + (−0.678 − 0.199i)2-s + (0.378 − 0.436i)3-s + (0.420 + 0.270i)4-s + (0.162 − 1.12i)5-s + (−0.343 + 0.220i)6-s + (−0.157 − 0.343i)7-s + (−0.231 − 0.267i)8-s + (−0.0474 − 0.329i)9-s + (−0.334 + 0.733i)10-s + (−1.09 + 0.320i)11-s + (0.276 − 0.0813i)12-s + (−0.504 + 1.10i)13-s + (0.0380 + 0.264i)14-s + (−0.430 − 0.497i)15-s + (0.103 + 0.227i)16-s + (−1.51 + 0.974i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.753 - 0.657i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.753 - 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0738883 + 0.197145i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0738883 + 0.197145i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.959 + 0.281i)T \) |
| 3 | \( 1 + (-0.654 + 0.755i)T \) |
| 7 | \( 1 + (0.415 + 0.909i)T \) |
| 23 | \( 1 + (0.557 + 4.76i)T \) |
| good | 5 | \( 1 + (-0.362 + 2.52i)T + (-4.79 - 1.40i)T^{2} \) |
| 11 | \( 1 + (3.62 - 1.06i)T + (9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (1.81 - 3.98i)T + (-8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (6.25 - 4.01i)T + (7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (5.28 + 3.39i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (4.07 - 2.62i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-6.48 - 7.48i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (-0.900 - 6.26i)T + (-35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (-0.830 + 5.77i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (-3.00 + 3.46i)T + (-6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 - 7.84T + 47T^{2} \) |
| 53 | \( 1 + (4.57 + 10.0i)T + (-34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (-1.22 + 2.67i)T + (-38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (5.35 + 6.18i)T + (-8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (7.27 + 2.13i)T + (56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (4.05 + 1.19i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (3.74 + 2.40i)T + (30.3 + 66.4i)T^{2} \) |
| 79 | \( 1 + (0.806 - 1.76i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (-1.52 - 10.5i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (-5.35 + 6.18i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (-1.00 + 7.00i)T + (-93.0 - 27.3i)T^{2} \) |
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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
−9.201392602322820043110708829723, −8.744877492340610332582771165426, −8.137652168353511605665228718441, −6.98731405678293872297343011122, −6.44728101635316846124565797541, −4.90710866945371928706708220293, −4.22478296518013897368448843285, −2.54491707240637508882122424039, −1.72789172228481931494173337988, −0.10487794473643730680306612093,
2.47531271437044640540144084343, 2.76866454510826908744143448358, 4.29257209121012869757208843588, 5.61587887646399940630171319174, 6.27348944280907507731825335922, 7.49038814048710261855664882255, 7.88404988336399209988807835186, 8.978288668854707303953851622537, 9.699029027736149087567867520309, 10.55157214993685805489029282502
