| L(s) = 1 | + (−1.20 + 0.747i)2-s + (0.644 + 1.60i)3-s + (0.881 − 1.79i)4-s + (3.32 + 0.585i)5-s + (−1.97 − 1.44i)6-s + (−1.72 − 2.05i)7-s + (0.285 + 2.81i)8-s + (−2.16 + 2.07i)9-s + (−4.42 + 1.78i)10-s + (0.506 + 2.87i)11-s + (3.45 + 0.259i)12-s + (0.552 − 0.201i)13-s + (3.61 + 1.17i)14-s + (1.19 + 5.71i)15-s + (−2.44 − 3.16i)16-s + (−6.36 − 3.67i)17-s + ⋯ |
| L(s) = 1 | + (−0.848 + 0.528i)2-s + (0.372 + 0.928i)3-s + (0.440 − 0.897i)4-s + (1.48 + 0.262i)5-s + (−0.806 − 0.591i)6-s + (−0.652 − 0.777i)7-s + (0.100 + 0.994i)8-s + (−0.723 + 0.690i)9-s + (−1.39 + 0.563i)10-s + (0.152 + 0.866i)11-s + (0.997 + 0.0749i)12-s + (0.153 − 0.0557i)13-s + (0.965 + 0.314i)14-s + (0.309 + 1.47i)15-s + (−0.611 − 0.791i)16-s + (−1.54 − 0.890i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.297 - 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.297 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.722650 + 0.531934i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.722650 + 0.531934i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.20 - 0.747i)T \) |
| 3 | \( 1 + (-0.644 - 1.60i)T \) |
| good | 5 | \( 1 + (-3.32 - 0.585i)T + (4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (1.72 + 2.05i)T + (-1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (-0.506 - 2.87i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (-0.552 + 0.201i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (6.36 + 3.67i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.82 + 1.63i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.365 - 0.306i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-1.49 + 4.10i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-3.73 + 4.45i)T + (-5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (1.59 - 2.76i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.09 + 3.01i)T + (-31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (1.79 - 0.316i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (0.940 - 0.789i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 - 7.37iT - 53T^{2} \) |
| 59 | \( 1 + (0.718 - 4.07i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-0.421 + 0.353i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (3.46 + 9.51i)T + (-51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (0.616 - 1.06i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.18 - 2.04i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3.33 - 9.16i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (16.8 + 6.15i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (4.81 - 2.77i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.83 - 10.4i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−13.88634890493993509366353777696, −13.52457880843652458974865284119, −11.28725143694624612171661760182, −10.16004911535749931050589140326, −9.737109437443052936008854188805, −8.913552983419003000691775197901, −7.22426556618135830651140410537, −6.17628327937766167887479047605, −4.75703880877912437108946959914, −2.45366194142412557905455544177,
1.72679897074524766963755363105, 3.00472154309855540433749899981, 5.94346068934688029368477542518, 6.73130026014049292978152716448, 8.555691836124369607825945941104, 8.980091072198626124229688542788, 10.10106613242468224227567721450, 11.42021539929533368830173758108, 12.63831820035056113313409489471, 13.19216985660394994615131022092
