| L(s) = 1 | + (−0.736 − 1.20i)2-s + (−0.195 + 0.980i)3-s + (−0.915 + 1.77i)4-s + (0.0836 + 0.0559i)5-s + (1.32 − 0.486i)6-s + (−4.46 + 1.85i)7-s + (2.82 − 0.203i)8-s + (−0.923 − 0.382i)9-s + (0.00589 − 0.142i)10-s + (−3.20 + 0.637i)11-s + (−1.56 − 1.24i)12-s + (−2.92 + 1.95i)13-s + (5.52 + 4.03i)14-s + (−0.0711 + 0.0711i)15-s + (−2.32 − 3.25i)16-s + (4.06 + 4.06i)17-s + ⋯ |
| L(s) = 1 | + (−0.520 − 0.853i)2-s + (−0.112 + 0.566i)3-s + (−0.457 + 0.889i)4-s + (0.0374 + 0.0250i)5-s + (0.542 − 0.198i)6-s + (−1.68 + 0.699i)7-s + (0.997 − 0.0720i)8-s + (−0.307 − 0.127i)9-s + (0.00186 − 0.0449i)10-s + (−0.965 + 0.192i)11-s + (−0.451 − 0.359i)12-s + (−0.812 + 0.542i)13-s + (1.47 + 1.07i)14-s + (−0.0183 + 0.0183i)15-s + (−0.580 − 0.814i)16-s + (0.985 + 0.985i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.239 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.239 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.246134 + 0.314375i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.246134 + 0.314375i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.736 + 1.20i)T \) |
| 3 | \( 1 + (0.195 - 0.980i)T \) |
| good | 5 | \( 1 + (-0.0836 - 0.0559i)T + (1.91 + 4.61i)T^{2} \) |
| 7 | \( 1 + (4.46 - 1.85i)T + (4.94 - 4.94i)T^{2} \) |
| 11 | \( 1 + (3.20 - 0.637i)T + (10.1 - 4.20i)T^{2} \) |
| 13 | \( 1 + (2.92 - 1.95i)T + (4.97 - 12.0i)T^{2} \) |
| 17 | \( 1 + (-4.06 - 4.06i)T + 17iT^{2} \) |
| 19 | \( 1 + (-1.61 - 2.41i)T + (-7.27 + 17.5i)T^{2} \) |
| 23 | \( 1 + (-1.02 + 2.46i)T + (-16.2 - 16.2i)T^{2} \) |
| 29 | \( 1 + (-2.77 - 0.551i)T + (26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 - 6.23iT - 31T^{2} \) |
| 37 | \( 1 + (3.81 - 5.71i)T + (-14.1 - 34.1i)T^{2} \) |
| 41 | \( 1 + (-3.15 + 7.61i)T + (-28.9 - 28.9i)T^{2} \) |
| 43 | \( 1 + (-0.370 - 1.86i)T + (-39.7 + 16.4i)T^{2} \) |
| 47 | \( 1 + (7.02 + 7.02i)T + 47iT^{2} \) |
| 53 | \( 1 + (4.73 - 0.941i)T + (48.9 - 20.2i)T^{2} \) |
| 59 | \( 1 + (-1.74 - 1.16i)T + (22.5 + 54.5i)T^{2} \) |
| 61 | \( 1 + (0.391 - 1.96i)T + (-56.3 - 23.3i)T^{2} \) |
| 67 | \( 1 + (2.53 - 12.7i)T + (-61.8 - 25.6i)T^{2} \) |
| 71 | \( 1 + (-0.0798 + 0.0330i)T + (50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (-3.83 - 1.58i)T + (51.6 + 51.6i)T^{2} \) |
| 79 | \( 1 + (-9.63 + 9.63i)T - 79iT^{2} \) |
| 83 | \( 1 + (-7.45 - 11.1i)T + (-31.7 + 76.6i)T^{2} \) |
| 89 | \( 1 + (-0.680 - 1.64i)T + (-62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 - 17.0iT - 97T^{2} \) |
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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
−12.33071330833922001155467225569, −12.15280980876531961331674588771, −10.37410227368794183390526342432, −10.11133673826267816827474842090, −9.166562055905187209932070021074, −8.107448302827380898588915075549, −6.63852661275535729579136627276, −5.21475534170653633442686018754, −3.64137538816208946639537787684, −2.59163491107704015412547233250,
0.40473053811623422501425095887, 3.07828824558731971550055566937, 5.13122337120727154673320869302, 6.15839654536642088418041390285, 7.29729461318960790809273946818, 7.74597390478287864245498866007, 9.463885949661678583192699736661, 9.873461829478469637390788595512, 11.06199730604598606118609931525, 12.60555865625923482882991830851
