| L(s) = 1 | + (2.14 − 0.378i)2-s + (−1.69 − 2.47i)3-s + (0.699 − 0.254i)4-s + (−1.51 − 1.80i)5-s + (−4.57 − 4.66i)6-s + (−3.52 − 6.09i)7-s + (−6.14 + 3.54i)8-s + (−3.24 + 8.39i)9-s + (−3.92 − 3.29i)10-s − 15.8i·11-s + (−1.81 − 1.29i)12-s + (0.0969 + 0.0813i)13-s + (−9.86 − 11.7i)14-s + (−1.89 + 6.79i)15-s + (−14.1 + 11.8i)16-s + (9.32 + 11.1i)17-s + ⋯ |
| L(s) = 1 | + (1.07 − 0.189i)2-s + (−0.565 − 0.824i)3-s + (0.174 − 0.0636i)4-s + (−0.302 − 0.360i)5-s + (−0.762 − 0.777i)6-s + (−0.503 − 0.871i)7-s + (−0.767 + 0.443i)8-s + (−0.360 + 0.932i)9-s + (−0.392 − 0.329i)10-s − 1.43i·11-s + (−0.151 − 0.108i)12-s + (0.00746 + 0.00625i)13-s + (−0.704 − 0.839i)14-s + (−0.126 + 0.453i)15-s + (−0.882 + 0.740i)16-s + (0.548 + 0.653i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.836 + 0.548i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.836 + 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.373819 - 1.25098i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.373819 - 1.25098i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.69 + 2.47i)T \) |
| 19 | \( 1 + (-10.8 + 15.6i)T \) |
| good | 2 | \( 1 + (-2.14 + 0.378i)T + (3.75 - 1.36i)T^{2} \) |
| 5 | \( 1 + (1.51 + 1.80i)T + (-4.34 + 24.6i)T^{2} \) |
| 7 | \( 1 + (3.52 + 6.09i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + 15.8iT - 121T^{2} \) |
| 13 | \( 1 + (-0.0969 - 0.0813i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (-9.32 - 11.1i)T + (-50.1 + 284. i)T^{2} \) |
| 23 | \( 1 + (5.73 + 15.7i)T + (-405. + 340. i)T^{2} \) |
| 29 | \( 1 + (-6.22 - 17.0i)T + (-644. + 540. i)T^{2} \) |
| 31 | \( 1 + 19.6T + 961T^{2} \) |
| 37 | \( 1 - 21.0T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-28.0 + 4.94i)T + (1.57e3 - 574. i)T^{2} \) |
| 43 | \( 1 + (-72.2 - 26.2i)T + (1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (15.6 + 42.8i)T + (-1.69e3 + 1.41e3i)T^{2} \) |
| 53 | \( 1 + (78.8 + 13.8i)T + (2.63e3 + 960. i)T^{2} \) |
| 59 | \( 1 + (26.4 - 72.6i)T + (-2.66e3 - 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-13.2 - 11.1i)T + (646. + 3.66e3i)T^{2} \) |
| 67 | \( 1 + (-8.03 + 45.5i)T + (-4.21e3 - 1.53e3i)T^{2} \) |
| 71 | \( 1 + (99.8 - 17.6i)T + (4.73e3 - 1.72e3i)T^{2} \) |
| 73 | \( 1 + (-32.7 - 11.9i)T + (4.08e3 + 3.42e3i)T^{2} \) |
| 79 | \( 1 + (-71.5 + 60.0i)T + (1.08e3 - 6.14e3i)T^{2} \) |
| 83 | \( 1 + (60.1 - 34.7i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (51.6 + 141. i)T + (-6.06e3 + 5.09e3i)T^{2} \) |
| 97 | \( 1 + (-21.4 - 121. i)T + (-8.84e3 + 3.21e3i)T^{2} \) |
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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
−12.41210501530974392484824487079, −11.42599733786784797676153289757, −10.60698005962820558701017027148, −8.873088197030133206695080083830, −7.82043834236114567482139657704, −6.47320190163390197603514810862, −5.60525085988984131895493600186, −4.32487709753697994280620180691, −3.03640250426938777843264706165, −0.61115138937614244782229543966,
3.07084916553574490698750313292, 4.21450543589233673125806391716, 5.31490527579593206583187329197, 6.10641218501697917471486470314, 7.42064573546527185554958200922, 9.413526253715784250351533497981, 9.687612269975887383193664037783, 11.22450902640926230387084341948, 12.22744266222151986089734779679, 12.62209324806933113975466105950
