L(s) = 1 | + (−0.654 − 0.755i)2-s + (−0.841 − 0.540i)3-s + (−0.142 + 0.989i)4-s + (1.60 − 3.51i)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 + (−3.70 + 1.08i)10-s + (−3.67 + 4.23i)11-s + (0.654 − 0.755i)12-s + (−5.89 + 1.73i)13-s + (−0.415 − 0.909i)14-s + (−3.25 + 2.08i)15-s + (−0.959 − 0.281i)16-s + (0.914 + 6.36i)17-s + ⋯ |
L(s) = 1 | + (−0.463 − 0.534i)2-s + (−0.485 − 0.312i)3-s + (−0.0711 + 0.494i)4-s + (0.717 − 1.57i)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 + (−1.17 + 0.344i)10-s + (−1.10 + 1.27i)11-s + (0.189 − 0.218i)12-s + (−1.63 + 0.480i)13-s + (−0.111 − 0.243i)14-s + (−0.839 + 0.539i)15-s + (−0.239 − 0.0704i)16-s + (0.221 + 1.54i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.106 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.106 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.187398 + 0.168437i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.187398 + 0.168437i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.23 - 4.63i)T \) |
good | 5 | \( 1 + (-1.60 + 3.51i)T + (-3.27 - 3.77i)T^{2} \) |
| 11 | \( 1 + (3.67 - 4.23i)T + (-1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (5.89 - 1.73i)T + (10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.914 - 6.36i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (-0.225 + 1.57i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (0.301 + 2.09i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (9.10 - 5.84i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (3.52 + 7.72i)T + (-24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (0.792 - 1.73i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (5.79 + 3.72i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 + (6.75 + 1.98i)T + (44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (2.86 - 0.841i)T + (49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (-1.93 + 1.24i)T + (25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (-4.84 - 5.59i)T + (-9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (1.63 + 1.88i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (0.628 - 4.36i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (2.07 - 0.608i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (-5.47 - 11.9i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (2.09 + 1.34i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-0.470 + 1.03i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−10.05464549103299421127267534033, −9.530390902992956852618092134408, −8.686899470663737261849930136179, −7.77422515795735736650507429020, −7.10073331288093225983042595261, −5.51142177319410866091811861134, −5.11509018543456652602721609857, −4.18417212936573197128174533742, −2.14431460200744659057199476509, −1.66473629202744493900152785268,
0.13392921682223506932129848946, 2.37904065501127433199338406874, 3.18925381348904889467685844984, 4.98456762453264842149519683955, 5.55862554795768296781255454386, 6.44100950416347652273116798911, 7.33901483270301580814663444918, 7.84043038158671219554337359982, 9.182889273035797724651280850803, 10.01490779198153215332949068844