| L(s) = 1 | + (0.327 + 0.945i)2-s + (−1.64 − 0.546i)3-s + (−0.786 + 0.618i)4-s + (−0.141 + 0.0415i)5-s + (−0.0208 − 1.73i)6-s + (−0.267 + 1.38i)7-s + (−0.841 − 0.540i)8-s + (2.40 + 1.79i)9-s + (−0.0855 − 0.120i)10-s + (−2.22 − 2.11i)11-s + (1.62 − 0.586i)12-s + (−4.74 + 0.225i)13-s + (−1.40 + 0.201i)14-s + (0.255 + 0.00908i)15-s + (0.235 − 0.971i)16-s + (−0.147 + 0.187i)17-s + ⋯ |
| L(s) = 1 | + (0.231 + 0.668i)2-s + (−0.948 − 0.315i)3-s + (−0.393 + 0.309i)4-s + (−0.0632 + 0.0185i)5-s + (−0.00849 − 0.707i)6-s + (−0.101 + 0.525i)7-s + (−0.297 − 0.191i)8-s + (0.800 + 0.599i)9-s + (−0.0270 − 0.0379i)10-s + (−0.669 − 0.638i)11-s + (0.470 − 0.169i)12-s + (−1.31 + 0.0626i)13-s + (−0.374 + 0.0538i)14-s + (0.0658 + 0.00234i)15-s + (0.0589 − 0.242i)16-s + (−0.0357 + 0.0455i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 402 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.851 + 0.524i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 402 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.851 + 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0279169 - 0.0986076i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0279169 - 0.0986076i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.327 - 0.945i)T \) |
| 3 | \( 1 + (1.64 + 0.546i)T \) |
| 67 | \( 1 + (-8.06 - 1.38i)T \) |
| good | 5 | \( 1 + (0.141 - 0.0415i)T + (4.20 - 2.70i)T^{2} \) |
| 7 | \( 1 + (0.267 - 1.38i)T + (-6.49 - 2.60i)T^{2} \) |
| 11 | \( 1 + (2.22 + 2.11i)T + (0.523 + 10.9i)T^{2} \) |
| 13 | \( 1 + (4.74 - 0.225i)T + (12.9 - 1.23i)T^{2} \) |
| 17 | \( 1 + (0.147 - 0.187i)T + (-4.00 - 16.5i)T^{2} \) |
| 19 | \( 1 + (6.09 - 1.17i)T + (17.6 - 7.06i)T^{2} \) |
| 23 | \( 1 + (-0.691 + 7.24i)T + (-22.5 - 4.35i)T^{2} \) |
| 29 | \( 1 + (2.94 - 1.69i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (1.81 + 0.0866i)T + (30.8 + 2.94i)T^{2} \) |
| 37 | \( 1 + (2.49 - 4.31i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-6.09 + 2.44i)T + (29.6 - 28.2i)T^{2} \) |
| 43 | \( 1 + (0.487 + 0.0701i)T + (41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + (2.08 + 1.48i)T + (15.3 + 44.4i)T^{2} \) |
| 53 | \( 1 + (-0.284 - 1.97i)T + (-50.8 + 14.9i)T^{2} \) |
| 59 | \( 1 + (6.16 - 9.59i)T + (-24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-4.78 - 5.01i)T + (-2.90 + 60.9i)T^{2} \) |
| 71 | \( 1 + (-2.44 - 3.11i)T + (-16.7 + 68.9i)T^{2} \) |
| 73 | \( 1 + (-0.428 + 0.408i)T + (3.47 - 72.9i)T^{2} \) |
| 79 | \( 1 + (-4.14 + 8.03i)T + (-45.8 - 64.3i)T^{2} \) |
| 83 | \( 1 + (17.3 + 4.20i)T + (73.7 + 38.0i)T^{2} \) |
| 89 | \( 1 + (-10.0 - 4.57i)T + (58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (13.9 + 8.04i)T + (48.5 + 84.0i)T^{2} \) |
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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
−12.01197762838927033895868285621, −10.91549241264329924146735674479, −10.12555434384438370525815948450, −8.884865819457645318434789441244, −7.891567709551619429314241257177, −6.98824930671606614497714785432, −6.04294439525187576062359240528, −5.28164287441772409686055283955, −4.27109424588225145787703311000, −2.42455301704032281513806138197,
0.06500409426220262596469317615, 2.12843219010602645748666290800, 3.84502888412118682334320086476, 4.73962141150499974495345623557, 5.61922103529920416613956387390, 6.88791494467344816922471799827, 7.80298271429851165143527830309, 9.455372696219927973715809412832, 9.965873683040187198061834509954, 10.82745017515689664686667206442
