| L(s) = 1 | + (0.0292 − 0.0551i)2-s + (−0.725 + 0.687i)3-s + (1.12 + 1.65i)4-s + (−2.41 + 0.531i)5-s + (0.0167 + 0.0601i)6-s + (−2.91 + 2.21i)7-s + (0.248 − 0.0269i)8-s + (0.0541 − 0.998i)9-s + (−0.0412 + 0.148i)10-s + (−1.21 − 3.04i)11-s + (−1.94 − 0.429i)12-s + (0.167 + 3.09i)13-s + (0.0370 + 0.225i)14-s + (1.38 − 2.04i)15-s + (−1.47 + 3.69i)16-s + (5.50 + 4.18i)17-s + ⋯ |
| L(s) = 1 | + (0.0206 − 0.0390i)2-s + (−0.419 + 0.397i)3-s + (0.560 + 0.826i)4-s + (−1.07 + 0.237i)5-s + (0.00682 + 0.0245i)6-s + (−1.10 + 0.838i)7-s + (0.0877 − 0.00954i)8-s + (0.0180 − 0.332i)9-s + (−0.0130 + 0.0470i)10-s + (−0.366 − 0.919i)11-s + (−0.562 − 0.123i)12-s + (0.0464 + 0.857i)13-s + (0.00989 + 0.0603i)14-s + (0.358 − 0.528i)15-s + (−0.367 + 0.923i)16-s + (1.33 + 1.01i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.505 - 0.863i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.505 - 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.382789 + 0.667531i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.382789 + 0.667531i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.725 - 0.687i)T \) |
| 59 | \( 1 + (2.54 + 7.24i)T \) |
| good | 2 | \( 1 + (-0.0292 + 0.0551i)T + (-1.12 - 1.65i)T^{2} \) |
| 5 | \( 1 + (2.41 - 0.531i)T + (4.53 - 2.09i)T^{2} \) |
| 7 | \( 1 + (2.91 - 2.21i)T + (1.87 - 6.74i)T^{2} \) |
| 11 | \( 1 + (1.21 + 3.04i)T + (-7.98 + 7.56i)T^{2} \) |
| 13 | \( 1 + (-0.167 - 3.09i)T + (-12.9 + 1.40i)T^{2} \) |
| 17 | \( 1 + (-5.50 - 4.18i)T + (4.54 + 16.3i)T^{2} \) |
| 19 | \( 1 + (-4.59 + 1.54i)T + (15.1 - 11.4i)T^{2} \) |
| 23 | \( 1 + (1.26 - 0.758i)T + (10.7 - 20.3i)T^{2} \) |
| 29 | \( 1 + (-0.671 - 1.26i)T + (-16.2 + 24.0i)T^{2} \) |
| 31 | \( 1 + (-5.47 - 1.84i)T + (24.6 + 18.7i)T^{2} \) |
| 37 | \( 1 + (0.515 + 0.0560i)T + (36.1 + 7.95i)T^{2} \) |
| 41 | \( 1 + (4.21 + 2.53i)T + (19.2 + 36.2i)T^{2} \) |
| 43 | \( 1 + (-3.09 + 7.77i)T + (-31.2 - 29.5i)T^{2} \) |
| 47 | \( 1 + (-0.114 - 0.0251i)T + (42.6 + 19.7i)T^{2} \) |
| 53 | \( 1 + (-1.48 - 5.36i)T + (-45.4 + 27.3i)T^{2} \) |
| 61 | \( 1 + (0.707 - 1.33i)T + (-34.2 - 50.4i)T^{2} \) |
| 67 | \( 1 + (-10.3 + 1.12i)T + (65.4 - 14.4i)T^{2} \) |
| 71 | \( 1 + (12.6 + 2.78i)T + (64.4 + 29.8i)T^{2} \) |
| 73 | \( 1 + (-2.54 - 15.5i)T + (-69.1 + 23.3i)T^{2} \) |
| 79 | \( 1 + (2.29 + 2.17i)T + (4.27 + 78.8i)T^{2} \) |
| 83 | \( 1 + (5.54 - 6.53i)T + (-13.4 - 81.9i)T^{2} \) |
| 89 | \( 1 + (-6.92 - 13.0i)T + (-49.9 + 73.6i)T^{2} \) |
| 97 | \( 1 + (-2.63 + 16.0i)T + (-91.9 - 30.9i)T^{2} \) |
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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
−12.58305732925448336766107083351, −11.97761441521794417396362925956, −11.34119106351966324991548445103, −10.19110317161599240727447834623, −8.886212823726938726008286461770, −7.895916375186870625870928333585, −6.76212553800168739037687562755, −5.68095847342299135604652571870, −3.80093082826408634118807789261, −3.06857951360744308401269021327,
0.74362353197878191192632759645, 3.15378768074521206608502276909, 4.81734959715189233137321528757, 6.05047892816193451196759950597, 7.28479431645074460199322376457, 7.73731646577430703011539084950, 9.851956703884105605691587745556, 10.20107332769483479342132800021, 11.56994965698090185529768430485, 12.18607941783984590416611483775
