| L(s) = 1 | + (−1.30 − 0.540i)2-s + (−1.71 + 0.234i)3-s + (1.41 + 1.41i)4-s + (0.137 − 0.0242i)5-s + (2.36 + 0.619i)6-s + (2.98 − 3.56i)7-s + (−1.08 − 2.61i)8-s + (2.88 − 0.806i)9-s + (−0.192 − 0.0424i)10-s + (0.182 − 1.03i)11-s + (−2.76 − 2.08i)12-s + (−1.46 − 0.532i)13-s + (−5.82 + 3.04i)14-s + (−0.229 + 0.0737i)15-s + (0.0146 + 3.99i)16-s + (5.25 − 3.03i)17-s + ⋯ |
| L(s) = 1 | + (−0.924 − 0.381i)2-s + (−0.990 + 0.135i)3-s + (0.708 + 0.705i)4-s + (0.0613 − 0.0108i)5-s + (0.967 + 0.252i)6-s + (1.12 − 1.34i)7-s + (−0.385 − 0.922i)8-s + (0.963 − 0.268i)9-s + (−0.0608 − 0.0134i)10-s + (0.0550 − 0.312i)11-s + (−0.797 − 0.603i)12-s + (−0.405 − 0.147i)13-s + (−1.55 + 0.812i)14-s + (−0.0593 + 0.0190i)15-s + (0.00365 + 0.999i)16-s + (1.27 − 0.736i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.575 + 0.817i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.575 + 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.520995 - 0.270350i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.520995 - 0.270350i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.30 + 0.540i)T \) |
| 3 | \( 1 + (1.71 - 0.234i)T \) |
| good | 5 | \( 1 + (-0.137 + 0.0242i)T + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-2.98 + 3.56i)T + (-1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (-0.182 + 1.03i)T + (-10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (1.46 + 0.532i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-5.25 + 3.03i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.80 - 2.19i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.94 - 1.62i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-0.210 - 0.578i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (3.18 + 3.79i)T + (-5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (1.43 + 2.49i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (2.25 - 6.18i)T + (-31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (2.15 + 0.380i)T + (40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-8.46 - 7.10i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + 8.73iT - 53T^{2} \) |
| 59 | \( 1 + (-1.42 - 8.08i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-8.56 - 7.18i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (1.93 - 5.30i)T + (-51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (2.33 + 4.04i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (5.64 - 9.77i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.572 + 1.57i)T + (-60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (2.20 - 0.802i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (-7.63 - 4.40i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.978 - 5.54i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−13.38317159503856788822041175047, −11.94331871881347655300605537038, −11.40501775873492759237502646837, −10.36966128124049036167282106349, −9.680604668856566336876054176488, −7.82645840436188847252683474941, −7.25931243404747831784029559845, −5.51460258533170937353403798074, −3.89486928710926192149035089001, −1.17609890307649109444026611313,
1.85147207871014782553842808381, 5.12576697235323039979259574000, 5.87622053401730347002706781519, 7.32017969992342736194979971150, 8.350972959392300427985516196853, 9.623331227782848631694484771940, 10.66098253743930245864349119025, 11.83619077271498303276945334390, 12.18768249284420063344496419370, 14.18331725165304810570403601601
