| 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
−10.55157214993685805489029282502, −9.699029027736149087567867520309, −8.978288668854707303953851622537, −7.88404988336399209988807835186, −7.49038814048710261855664882255, −6.27348944280907507731825335922, −5.61587887646399940630171319174, −4.29257209121012869757208843588, −2.76866454510826908744143448358, −2.47531271437044640540144084343,
0.10487794473643730680306612093, 1.72789172228481931494173337988, 2.54491707240637508882122424039, 4.22478296518013897368448843285, 4.90710866945371928706708220293, 6.44728101635316846124565797541, 6.98731405678293872297343011122, 8.137652168353511605665228718441, 8.744877492340610332582771165426, 9.201392602322820043110708829723
