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 | \( 1 + (-2.63 - 16.0i)T + (-91.9 + 30.9i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.18607941783984590416611483775, −11.56994965698090185529768430485, −10.20107332769483479342132800021, −9.851956703884105605691587745556, −7.73731646577430703011539084950, −7.28479431645074460199322376457, −6.05047892816193451196759950597, −4.81734959715189233137321528757, −3.15378768074521206608502276909, −0.74362353197878191192632759645,
3.06857951360744308401269021327, 3.80093082826408634118807789261, 5.68095847342299135604652571870, 6.76212553800168739037687562755, 7.895916375186870625870928333585, 8.886212823726938726008286461770, 10.19110317161599240727447834623, 11.34119106351966324991548445103, 11.97761441521794417396362925956, 12.58305732925448336766107083351