| L(s) = 1 | + (0.359 − 1.36i)2-s + (0.298 − 1.70i)3-s + (−1.74 − 0.983i)4-s + (0.924 + 1.05i)5-s + (−2.22 − 1.02i)6-s + (4.48 + 0.590i)7-s + (−1.97 + 2.02i)8-s + (−2.82 − 1.01i)9-s + (1.77 − 0.885i)10-s + (−1.71 − 3.47i)11-s + (−2.19 + 2.67i)12-s + (−0.777 − 2.29i)13-s + (2.41 − 5.91i)14-s + (2.07 − 1.26i)15-s + (2.06 + 3.42i)16-s + (−4.38 − 4.38i)17-s + ⋯ |
| L(s) = 1 | + (0.254 − 0.967i)2-s + (0.172 − 0.985i)3-s + (−0.870 − 0.491i)4-s + (0.413 + 0.471i)5-s + (−0.908 − 0.417i)6-s + (1.69 + 0.223i)7-s + (−0.697 + 0.716i)8-s + (−0.940 − 0.339i)9-s + (0.560 − 0.279i)10-s + (−0.516 − 1.04i)11-s + (−0.634 + 0.772i)12-s + (−0.215 − 0.635i)13-s + (0.646 − 1.58i)14-s + (0.535 − 0.325i)15-s + (0.516 + 0.856i)16-s + (−1.06 − 1.06i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.897 + 0.440i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.897 + 0.440i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.401329 - 1.72831i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.401329 - 1.72831i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.359 + 1.36i)T \) |
| 3 | \( 1 + (-0.298 + 1.70i)T \) |
| good | 5 | \( 1 + (-0.924 - 1.05i)T + (-0.652 + 4.95i)T^{2} \) |
| 7 | \( 1 + (-4.48 - 0.590i)T + (6.76 + 1.81i)T^{2} \) |
| 11 | \( 1 + (1.71 + 3.47i)T + (-6.69 + 8.72i)T^{2} \) |
| 13 | \( 1 + (0.777 + 2.29i)T + (-10.3 + 7.91i)T^{2} \) |
| 17 | \( 1 + (4.38 + 4.38i)T + 17iT^{2} \) |
| 19 | \( 1 + (-0.960 + 4.82i)T + (-17.5 - 7.27i)T^{2} \) |
| 23 | \( 1 + (-1.02 - 7.76i)T + (-22.2 + 5.95i)T^{2} \) |
| 29 | \( 1 + (-7.06 + 0.462i)T + (28.7 - 3.78i)T^{2} \) |
| 31 | \( 1 + (0.129 - 0.0748i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.836 - 4.20i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (-0.688 - 5.22i)T + (-39.6 + 10.6i)T^{2} \) |
| 43 | \( 1 + (0.680 - 0.335i)T + (26.1 - 34.1i)T^{2} \) |
| 47 | \( 1 + (6.96 - 1.86i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-2.43 + 3.64i)T + (-20.2 - 48.9i)T^{2} \) |
| 59 | \( 1 + (3.61 + 4.12i)T + (-7.70 + 58.4i)T^{2} \) |
| 61 | \( 1 + (-0.273 - 4.16i)T + (-60.4 + 7.96i)T^{2} \) |
| 67 | \( 1 + (-14.0 - 6.93i)T + (40.7 + 53.1i)T^{2} \) |
| 71 | \( 1 + (-4.52 - 10.9i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (-2.67 + 6.45i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (-0.189 - 0.705i)T + (-68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (-3.58 - 3.14i)T + (10.8 + 82.2i)T^{2} \) |
| 89 | \( 1 + (3.19 - 1.32i)T + (62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 + (-10.7 - 6.19i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.74674308030563951719332339625, −9.505010962141351365164208533328, −8.490506920500698085215427275060, −7.967148451399721563626588411280, −6.68985014250697775305492683597, −5.47387296278670663745029281414, −4.84163845145662668465344434099, −3.00102020581382412742436636557, −2.33035333463170034836823054263, −0.975532011311052411760550248048,
2.07394072227110035240241488094, 4.06046411996109193302923942504, 4.70392215165789355230529308465, 5.24270863822987432336313421977, 6.48903456006182589097606009786, 7.75620432773842942861192176152, 8.425858470541264766350620960543, 9.080452050489489480790534849558, 10.15456390077126965357433560565, 10.85721419910210005590939547965
