| L(s) = 1 | + (0.415 + 0.909i)2-s + (0.959 − 0.281i)3-s + (−0.654 + 0.755i)4-s + (−3.48 − 2.23i)5-s + (0.654 + 0.755i)6-s + (0.142 + 0.989i)7-s + (−0.959 − 0.281i)8-s + (0.841 − 0.540i)9-s + (0.589 − 4.10i)10-s + (0.455 − 0.998i)11-s + (−0.415 + 0.909i)12-s + (−0.581 + 4.04i)13-s + (−0.841 + 0.540i)14-s + (−3.97 − 1.16i)15-s + (−0.142 − 0.989i)16-s + (4.67 + 5.39i)17-s + ⋯ |
| L(s) = 1 | + (0.293 + 0.643i)2-s + (0.553 − 0.162i)3-s + (−0.327 + 0.377i)4-s + (−1.55 − 1.00i)5-s + (0.267 + 0.308i)6-s + (0.0537 + 0.374i)7-s + (−0.339 − 0.0996i)8-s + (0.280 − 0.180i)9-s + (0.186 − 1.29i)10-s + (0.137 − 0.301i)11-s + (−0.119 + 0.262i)12-s + (−0.161 + 1.12i)13-s + (−0.224 + 0.144i)14-s + (−1.02 − 0.301i)15-s + (−0.0355 − 0.247i)16-s + (1.13 + 1.30i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.350 - 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.350 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.737231 + 1.06352i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.737231 + 1.06352i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.415 - 0.909i)T \) |
| 3 | \( 1 + (-0.959 + 0.281i)T \) |
| 7 | \( 1 + (-0.142 - 0.989i)T \) |
| 23 | \( 1 + (-2.97 - 3.76i)T \) |
| good | 5 | \( 1 + (3.48 + 2.23i)T + (2.07 + 4.54i)T^{2} \) |
| 11 | \( 1 + (-0.455 + 0.998i)T + (-7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (0.581 - 4.04i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (-4.67 - 5.39i)T + (-2.41 + 16.8i)T^{2} \) |
| 19 | \( 1 + (4.33 - 4.99i)T + (-2.70 - 18.8i)T^{2} \) |
| 29 | \( 1 + (1.47 + 1.70i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (-7.61 - 2.23i)T + (26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (1.40 - 0.902i)T + (15.3 - 33.6i)T^{2} \) |
| 41 | \( 1 + (-0.991 - 0.637i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (3.74 - 1.09i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + 3.81T + 47T^{2} \) |
| 53 | \( 1 + (1.68 + 11.7i)T + (-50.8 + 14.9i)T^{2} \) |
| 59 | \( 1 + (0.203 - 1.41i)T + (-56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (4.03 + 1.18i)T + (51.3 + 32.9i)T^{2} \) |
| 67 | \( 1 + (-1.61 - 3.52i)T + (-43.8 + 50.6i)T^{2} \) |
| 71 | \( 1 + (-2.74 - 6.01i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (8.19 - 9.46i)T + (-10.3 - 72.2i)T^{2} \) |
| 79 | \( 1 + (-2.11 + 14.7i)T + (-75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (11.4 - 7.32i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (-10.6 + 3.13i)T + (74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (-1.07 - 0.690i)T + (40.2 + 88.2i)T^{2} \) |
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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
−10.06409839133851676719446954129, −9.026203136093313540407883342747, −8.261225584054420824585890819393, −8.085802546504827109339883347999, −7.03871411287405452763028529543, −6.02173319732488588785904711835, −4.88464595772006702432011738501, −4.03060214556942056798075879657, −3.41876813767393504308986597607, −1.48200117331904934140902751080,
0.54774968262389248701140513877, 2.76501104322667668087080587153, 3.13876214460563586799368880268, 4.23699061520227925700355945662, 4.93379882044821607306620258486, 6.56050454555400515261975965437, 7.39998138596303125628551450241, 7.981743636934776428199582976864, 8.948926647869793658011915468124, 10.06381032795671188594356481815
