L(s) = 1 | + (0.811 + 0.318i)2-s + (−0.908 − 0.843i)4-s + (−0.228 − 0.155i)5-s + (−2.58 + 0.584i)7-s + (−1.22 − 2.54i)8-s + (−0.135 − 0.199i)10-s + (−0.640 − 4.25i)11-s + (3.73 − 2.98i)13-s + (−2.28 − 0.347i)14-s + (0.00131 + 0.0176i)16-s + (−5.10 − 1.57i)17-s + (1.61 − 0.930i)19-s + (0.0764 + 0.334i)20-s + (0.833 − 3.65i)22-s + (−0.312 − 1.01i)23-s + ⋯ |
L(s) = 1 | + (0.573 + 0.225i)2-s + (−0.454 − 0.421i)4-s + (−0.102 − 0.0697i)5-s + (−0.975 + 0.220i)7-s + (−0.433 − 0.899i)8-s + (−0.0429 − 0.0630i)10-s + (−0.193 − 1.28i)11-s + (1.03 − 0.826i)13-s + (−0.609 − 0.0928i)14-s + (0.000329 + 0.00440i)16-s + (−1.23 − 0.381i)17-s + (0.369 − 0.213i)19-s + (0.0170 + 0.0748i)20-s + (0.177 − 0.778i)22-s + (−0.0652 − 0.211i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.658313 - 0.831931i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.658313 - 0.831931i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (2.58 - 0.584i)T \) |
good | 2 | \( 1 + (-0.811 - 0.318i)T + (1.46 + 1.36i)T^{2} \) |
| 5 | \( 1 + (0.228 + 0.155i)T + (1.82 + 4.65i)T^{2} \) |
| 11 | \( 1 + (0.640 + 4.25i)T + (-10.5 + 3.24i)T^{2} \) |
| 13 | \( 1 + (-3.73 + 2.98i)T + (2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (5.10 + 1.57i)T + (14.0 + 9.57i)T^{2} \) |
| 19 | \( 1 + (-1.61 + 0.930i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.312 + 1.01i)T + (-19.0 + 12.9i)T^{2} \) |
| 29 | \( 1 + (3.23 - 0.737i)T + (26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (-4.59 - 2.65i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (5.88 - 5.46i)T + (2.76 - 36.8i)T^{2} \) |
| 41 | \( 1 + (5.58 - 2.69i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (-9.74 - 4.69i)T + (26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (-3.78 + 9.64i)T + (-34.4 - 31.9i)T^{2} \) |
| 53 | \( 1 + (2.18 - 2.35i)T + (-3.96 - 52.8i)T^{2} \) |
| 59 | \( 1 + (-10.0 + 6.84i)T + (21.5 - 54.9i)T^{2} \) |
| 61 | \( 1 + (-5.45 - 5.88i)T + (-4.55 + 60.8i)T^{2} \) |
| 67 | \( 1 + (-3.64 + 6.32i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (4.23 + 0.965i)T + (63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (4.31 - 1.69i)T + (53.5 - 49.6i)T^{2} \) |
| 79 | \( 1 + (-1.86 - 3.22i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.641 + 0.804i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (-15.0 - 2.26i)T + (85.0 + 26.2i)T^{2} \) |
| 97 | \( 1 + 10.6iT - 97T^{2} \) |
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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
−10.79010456558698898637088171759, −10.01560469970202252335487564220, −8.941722720262190534048398989766, −8.370030136928725556564872491607, −6.72875356684772366950340111749, −6.07354188066054408685830722503, −5.22445225651178236175350945351, −3.92110493025536942643114820045, −2.99378002275762709903503769745, −0.55469036479192190586821792482,
2.22761943252480186919636129806, 3.69958819430875747291967957520, 4.25567517635804196766275160950, 5.57647835412087185224920899166, 6.70880595102392004022658499594, 7.60822867457787825230860182538, 8.889762807307951660850466997109, 9.431072773541281679246551006522, 10.58620725217341576580703585541, 11.56585621575578244912998805827