| L(s) = 1 | + (1.13 + 0.849i)2-s + (−0.881 + 0.471i)3-s + (0.557 + 1.92i)4-s + (−2.52 − 3.07i)5-s + (−1.39 − 0.215i)6-s + (−4.25 + 2.84i)7-s + (−0.999 + 2.64i)8-s + (0.555 − 0.831i)9-s + (−0.243 − 5.62i)10-s + (2.06 − 0.625i)11-s + (−1.39 − 1.43i)12-s + (−4.57 − 3.75i)13-s + (−7.22 − 0.397i)14-s + (3.67 + 1.52i)15-s + (−3.37 + 2.14i)16-s + (−0.968 + 0.400i)17-s + ⋯ |
| L(s) = 1 | + (0.799 + 0.600i)2-s + (−0.509 + 0.272i)3-s + (0.278 + 0.960i)4-s + (−1.12 − 1.37i)5-s + (−0.570 − 0.0880i)6-s + (−1.60 + 1.07i)7-s + (−0.353 + 0.935i)8-s + (0.185 − 0.277i)9-s + (−0.0768 − 1.77i)10-s + (0.621 − 0.188i)11-s + (−0.403 − 0.413i)12-s + (−1.26 − 1.04i)13-s + (−1.93 − 0.106i)14-s + (0.949 + 0.393i)15-s + (−0.844 + 0.535i)16-s + (−0.234 + 0.0972i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.853 + 0.520i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.853 + 0.520i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0592614 - 0.210942i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0592614 - 0.210942i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.13 - 0.849i)T \) |
| 3 | \( 1 + (0.881 - 0.471i)T \) |
| good | 5 | \( 1 + (2.52 + 3.07i)T + (-0.975 + 4.90i)T^{2} \) |
| 7 | \( 1 + (4.25 - 2.84i)T + (2.67 - 6.46i)T^{2} \) |
| 11 | \( 1 + (-2.06 + 0.625i)T + (9.14 - 6.11i)T^{2} \) |
| 13 | \( 1 + (4.57 + 3.75i)T + (2.53 + 12.7i)T^{2} \) |
| 17 | \( 1 + (0.968 - 0.400i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (-0.230 - 2.34i)T + (-18.6 + 3.70i)T^{2} \) |
| 23 | \( 1 + (0.902 - 0.179i)T + (21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (1.08 - 3.57i)T + (-24.1 - 16.1i)T^{2} \) |
| 31 | \( 1 + (1.11 + 1.11i)T + 31iT^{2} \) |
| 37 | \( 1 + (-2.94 - 0.289i)T + (36.2 + 7.21i)T^{2} \) |
| 41 | \( 1 + (-1.55 - 7.81i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (4.62 + 2.47i)T + (23.8 + 35.7i)T^{2} \) |
| 47 | \( 1 + (-0.425 - 1.02i)T + (-33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (0.296 + 0.976i)T + (-44.0 + 29.4i)T^{2} \) |
| 59 | \( 1 + (-3.47 + 2.85i)T + (11.5 - 57.8i)T^{2} \) |
| 61 | \( 1 + (-3.89 - 7.28i)T + (-33.8 + 50.7i)T^{2} \) |
| 67 | \( 1 + (1.86 + 3.48i)T + (-37.2 + 55.7i)T^{2} \) |
| 71 | \( 1 + (-2.40 - 3.59i)T + (-27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (10.9 + 7.31i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (5.55 - 13.4i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (-3.28 + 0.323i)T + (81.4 - 16.1i)T^{2} \) |
| 89 | \( 1 + (13.9 + 2.78i)T + (82.2 + 34.0i)T^{2} \) |
| 97 | \( 1 + (6.06 + 6.06i)T + 97iT^{2} \) |
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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
−12.15137855246548554233138223178, −11.52050353357265099724352375242, −9.847749924676235484317340190692, −8.980000554170008662738729397132, −8.129094697618666763999368765099, −7.01124342285181573402575066806, −5.87181493539551161285914064401, −5.17665184617149307264343087025, −4.08550158103313660610571271273, −3.06673623352565439671857825980,
0.11204114890133790447117725476, 2.60875863876806789539882720870, 3.71942139488789949713717092148, 4.40594735946451114815278507106, 6.23689762813766850440116772185, 7.01389714163847911077920483744, 7.20817789019782645894278888380, 9.498566008453702817679146451639, 10.17501602521640665219168469570, 11.00604542030129682801515070863
