| L(s) = 1 | + (0.854 + 1.12i)2-s + (0.00846 + 0.0480i)3-s + (−0.539 + 1.92i)4-s + (0.579 − 0.486i)5-s + (−0.0468 + 0.0505i)6-s + (−2.62 − 1.51i)7-s + (−2.63 + 1.03i)8-s + (2.81 − 1.02i)9-s + (1.04 + 0.237i)10-s + (0.655 − 0.378i)11-s + (−0.0970 − 0.00958i)12-s + (−1.53 − 0.270i)13-s + (−0.536 − 4.25i)14-s + (0.0282 + 0.0237i)15-s + (−3.41 − 2.07i)16-s + (−4.84 − 1.76i)17-s + ⋯ |
| L(s) = 1 | + (0.604 + 0.796i)2-s + (0.00488 + 0.0277i)3-s + (−0.269 + 0.962i)4-s + (0.259 − 0.217i)5-s + (−0.0191 + 0.0206i)6-s + (−0.992 − 0.573i)7-s + (−0.930 + 0.367i)8-s + (0.938 − 0.341i)9-s + (0.330 + 0.0751i)10-s + (0.197 − 0.114i)11-s + (−0.0280 − 0.00276i)12-s + (−0.425 − 0.0750i)13-s + (−0.143 − 1.13i)14-s + (0.00730 + 0.00612i)15-s + (−0.854 − 0.519i)16-s + (−1.17 − 0.427i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.537 - 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.537 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.02586 + 0.562818i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.02586 + 0.562818i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.854 - 1.12i)T \) |
| 19 | \( 1 + (-4.29 - 0.753i)T \) |
| good | 3 | \( 1 + (-0.00846 - 0.0480i)T + (-2.81 + 1.02i)T^{2} \) |
| 5 | \( 1 + (-0.579 + 0.486i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (2.62 + 1.51i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.655 + 0.378i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.53 + 0.270i)T + (12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (4.84 + 1.76i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (-1.13 + 1.34i)T + (-3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (0.178 + 0.491i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (3.59 - 6.23i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 8.80iT - 37T^{2} \) |
| 41 | \( 1 + (2.56 - 0.452i)T + (38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-6.43 - 7.66i)T + (-7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (3.62 + 9.96i)T + (-36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (-5.23 + 6.24i)T + (-9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (-7.79 - 2.83i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (2.83 + 2.38i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-8.84 + 3.21i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (6.90 - 5.79i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (1.69 + 9.64i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (2.62 + 14.9i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (10.6 + 6.15i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-13.0 - 2.30i)T + (83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (1.74 - 4.80i)T + (-74.3 - 62.3i)T^{2} \) |
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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
−14.73779616071892451753064679408, −13.43039296932022441996139874541, −13.00337887193459378002978841123, −11.75668195188951287953867922169, −9.994627939804260840803930555776, −9.007480004963884059104772385710, −7.30145551363217792960821244596, −6.54167363808991066025855764567, −4.91742577010965274412773837365, −3.47646788702722401314722244018,
2.38919251331963762321589446554, 4.12695703356711556144873127713, 5.73377026668083740700600574798, 7.00521900795031332449803759266, 9.182757788742355631953340106581, 9.951562248413772815150815532117, 11.11166365579664657975016414193, 12.40669245715507266255309153988, 13.06852542841963368153786016184, 14.08421138811894675071260303972
