| L(s) = 1 | + (1.94 − 0.938i)2-s + (−2.41 + 1.64i)3-s + (1.67 − 2.09i)4-s + (−2.00 − 1.85i)5-s + (−3.15 + 5.46i)6-s + (1.01 + 1.75i)7-s + (0.326 − 1.43i)8-s + (2.01 − 5.13i)9-s + (−5.64 − 1.74i)10-s + (−0.515 − 0.646i)11-s + (−0.584 + 7.79i)12-s + (3.02 − 0.931i)13-s + (3.61 + 2.46i)14-s + (7.87 + 1.18i)15-s + (0.485 + 2.12i)16-s + (−2.50 + 2.32i)17-s + ⋯ |
| L(s) = 1 | + (1.37 − 0.663i)2-s + (−1.39 + 0.949i)3-s + (0.835 − 1.04i)4-s + (−0.894 − 0.830i)5-s + (−1.28 + 2.23i)6-s + (0.382 + 0.661i)7-s + (0.115 − 0.505i)8-s + (0.671 − 1.71i)9-s + (−1.78 − 0.550i)10-s + (−0.155 − 0.194i)11-s + (−0.168 + 2.25i)12-s + (0.837 − 0.258i)13-s + (0.965 + 0.658i)14-s + (2.03 + 0.306i)15-s + (0.121 + 0.531i)16-s + (−0.607 + 0.563i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.315i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.949 + 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.962524 - 0.155693i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.962524 - 0.155693i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (3.02 - 5.82i)T \) |
| good | 2 | \( 1 + (-1.94 + 0.938i)T + (1.24 - 1.56i)T^{2} \) |
| 3 | \( 1 + (2.41 - 1.64i)T + (1.09 - 2.79i)T^{2} \) |
| 5 | \( 1 + (2.00 + 1.85i)T + (0.373 + 4.98i)T^{2} \) |
| 7 | \( 1 + (-1.01 - 1.75i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.515 + 0.646i)T + (-2.44 + 10.7i)T^{2} \) |
| 13 | \( 1 + (-3.02 + 0.931i)T + (10.7 - 7.32i)T^{2} \) |
| 17 | \( 1 + (2.50 - 2.32i)T + (1.27 - 16.9i)T^{2} \) |
| 19 | \( 1 + (1.08 + 2.75i)T + (-13.9 + 12.9i)T^{2} \) |
| 23 | \( 1 + (-1.21 + 0.182i)T + (21.9 - 6.77i)T^{2} \) |
| 29 | \( 1 + (7.00 + 4.77i)T + (10.5 + 26.9i)T^{2} \) |
| 31 | \( 1 + (0.399 - 5.32i)T + (-30.6 - 4.62i)T^{2} \) |
| 37 | \( 1 + (1.73 - 3.00i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-6.20 + 2.98i)T + (25.5 - 32.0i)T^{2} \) |
| 47 | \( 1 + (-2.33 + 2.93i)T + (-10.4 - 45.8i)T^{2} \) |
| 53 | \( 1 + (-8.47 - 2.61i)T + (43.7 + 29.8i)T^{2} \) |
| 59 | \( 1 + (0.633 + 2.77i)T + (-53.1 + 25.5i)T^{2} \) |
| 61 | \( 1 + (-0.573 - 7.65i)T + (-60.3 + 9.09i)T^{2} \) |
| 67 | \( 1 + (3.76 + 9.59i)T + (-49.1 + 45.5i)T^{2} \) |
| 71 | \( 1 + (11.9 + 1.80i)T + (67.8 + 20.9i)T^{2} \) |
| 73 | \( 1 + (-11.9 + 3.67i)T + (60.3 - 41.1i)T^{2} \) |
| 79 | \( 1 + (2.34 + 4.05i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (3.94 - 2.68i)T + (30.3 - 77.2i)T^{2} \) |
| 89 | \( 1 + (1.70 - 1.16i)T + (32.5 - 82.8i)T^{2} \) |
| 97 | \( 1 + (7.51 + 9.42i)T + (-21.5 + 94.5i)T^{2} \) |
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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
−15.62908523267559272990347595814, −15.12725938447865338016237968196, −13.21310185251940928695086204828, −12.19889221304371525702032022126, −11.44748740649641059527574334979, −10.71864914382789654180738355826, −8.702572675021403396711134893494, −5.95894704839175781782770554727, −4.94173757451456516646385547358, −3.97146891965365363225326055900,
4.08420244701254745454034190579, 5.65624958057613059112437275403, 6.89312978465759841263948405853, 7.50651176955025974630146498929, 10.90260936019483346597097724797, 11.55255582210315796889245628766, 12.70464877219228940132302930824, 13.64850550367894038280092405265, 14.83469822843574019246707509124, 15.98468301617368715791202421168
