| L(s) = 1 | + (0.192 − 1.40i)2-s + (0.881 − 0.471i)3-s + (−1.92 − 0.539i)4-s + (−1.13 − 1.37i)5-s + (−0.490 − 1.32i)6-s + (−3.27 + 2.19i)7-s + (−1.12 + 2.59i)8-s + (0.555 − 0.831i)9-s + (−2.15 + 1.32i)10-s + (−3.58 + 1.08i)11-s + (−1.95 + 0.432i)12-s + (−5.07 − 4.16i)13-s + (2.43 + 5.01i)14-s + (−1.64 − 0.682i)15-s + (3.41 + 2.07i)16-s + (3.69 − 1.53i)17-s + ⋯ |
| L(s) = 1 | + (0.136 − 0.990i)2-s + (0.509 − 0.272i)3-s + (−0.962 − 0.269i)4-s + (−0.506 − 0.616i)5-s + (−0.200 − 0.541i)6-s + (−1.23 + 0.827i)7-s + (−0.398 + 0.917i)8-s + (0.185 − 0.277i)9-s + (−0.680 + 0.417i)10-s + (−1.07 + 0.327i)11-s + (−0.563 + 0.124i)12-s + (−1.40 − 1.15i)13-s + (0.651 + 1.34i)14-s + (−0.425 − 0.176i)15-s + (0.854 + 0.519i)16-s + (0.895 − 0.371i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.818 - 0.574i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.818 - 0.574i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.171452 + 0.542497i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.171452 + 0.542497i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.192 + 1.40i)T \) |
| 3 | \( 1 + (-0.881 + 0.471i)T \) |
| good | 5 | \( 1 + (1.13 + 1.37i)T + (-0.975 + 4.90i)T^{2} \) |
| 7 | \( 1 + (3.27 - 2.19i)T + (2.67 - 6.46i)T^{2} \) |
| 11 | \( 1 + (3.58 - 1.08i)T + (9.14 - 6.11i)T^{2} \) |
| 13 | \( 1 + (5.07 + 4.16i)T + (2.53 + 12.7i)T^{2} \) |
| 17 | \( 1 + (-3.69 + 1.53i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (0.395 + 4.02i)T + (-18.6 + 3.70i)T^{2} \) |
| 23 | \( 1 + (-4.46 + 0.888i)T + (21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (0.526 - 1.73i)T + (-24.1 - 16.1i)T^{2} \) |
| 31 | \( 1 + (0.107 + 0.107i)T + 31iT^{2} \) |
| 37 | \( 1 + (5.97 + 0.588i)T + (36.2 + 7.21i)T^{2} \) |
| 41 | \( 1 + (0.531 + 2.67i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (-6.31 - 3.37i)T + (23.8 + 35.7i)T^{2} \) |
| 47 | \( 1 + (2.41 + 5.83i)T + (-33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (-2.32 - 7.65i)T + (-44.0 + 29.4i)T^{2} \) |
| 59 | \( 1 + (8.94 - 7.34i)T + (11.5 - 57.8i)T^{2} \) |
| 61 | \( 1 + (7.19 + 13.4i)T + (-33.8 + 50.7i)T^{2} \) |
| 67 | \( 1 + (1.25 + 2.35i)T + (-37.2 + 55.7i)T^{2} \) |
| 71 | \( 1 + (-5.92 - 8.86i)T + (-27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (-3.82 - 2.55i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (0.938 - 2.26i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (2.92 - 0.288i)T + (81.4 - 16.1i)T^{2} \) |
| 89 | \( 1 + (-8.98 - 1.78i)T + (82.2 + 34.0i)T^{2} \) |
| 97 | \( 1 + (11.0 + 11.0i)T + 97iT^{2} \) |
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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
−10.70313752695327412706123331599, −9.831595208363153520577817997721, −9.168709032539976400432744880168, −8.225711734938810334334981966735, −7.24889198217797676112231272753, −5.54977457186345692652130156026, −4.78215353289513705277311922853, −3.11934621783292926750177981587, −2.60296093220267965987098492295, −0.32608623990050110758379766751,
3.06722276079270279169897339951, 3.85766548325837938157444263631, 5.08640599647340777159540917122, 6.41665783009069042489063260087, 7.34653983101406760872061360944, 7.77143803398566982228744044787, 9.143606262643369894202357612299, 9.911381620975105079984702287298, 10.60989022277253460619907280003, 12.14713342339368874524967209924
