| 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
−14.08421138811894675071260303972, −13.06852542841963368153786016184, −12.40669245715507266255309153988, −11.11166365579664657975016414193, −9.951562248413772815150815532117, −9.182757788742355631953340106581, −7.00521900795031332449803759266, −5.73377026668083740700600574798, −4.12695703356711556144873127713, −2.38919251331963762321589446554,
3.47646788702722401314722244018, 4.91742577010965274412773837365, 6.54167363808991066025855764567, 7.30145551363217792960821244596, 9.007480004963884059104772385710, 9.994627939804260840803930555776, 11.75668195188951287953867922169, 13.00337887193459378002978841123, 13.43039296932022441996139874541, 14.73779616071892451753064679408
