L(s) = 1 | + (1.02 + 1.77i)2-s + (−1.10 + 1.92i)4-s − 0.146·5-s − 0.446·8-s + (−0.150 − 0.260i)10-s − 1.66·11-s + (−0.0999 − 0.173i)13-s + (1.75 + 3.04i)16-s + (3.13 + 5.43i)17-s + (−3.45 + 5.99i)19-s + (0.162 − 0.280i)20-s + (−1.70 − 2.95i)22-s + 6.18·23-s − 4.97·25-s + (0.205 − 0.355i)26-s + ⋯ |
L(s) = 1 | + (0.726 + 1.25i)2-s + (−0.554 + 0.960i)4-s − 0.0654·5-s − 0.157·8-s + (−0.0474 − 0.0822i)10-s − 0.501·11-s + (−0.0277 − 0.0480i)13-s + (0.439 + 0.761i)16-s + (0.760 + 1.31i)17-s + (−0.793 + 1.37i)19-s + (0.0362 − 0.0627i)20-s + (−0.364 − 0.630i)22-s + 1.28·23-s − 0.995·25-s + (0.0402 − 0.0697i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.893 - 0.449i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.893 - 0.449i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.228910811\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.228910811\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-1.02 - 1.77i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + 0.146T + 5T^{2} \) |
| 11 | \( 1 + 1.66T + 11T^{2} \) |
| 13 | \( 1 + (0.0999 + 0.173i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.13 - 5.43i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.45 - 5.99i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 6.18T + 23T^{2} \) |
| 29 | \( 1 + (-2.46 + 4.27i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (1.25 - 2.18i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.50 - 6.06i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.15 - 2.00i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.940 - 1.62i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.905 - 1.56i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.67 - 4.62i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.28 + 3.95i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (0.339 + 0.587i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.09 + 5.35i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 1.27T + 71T^{2} \) |
| 73 | \( 1 + (-0.778 - 1.34i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (6.39 + 11.0i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.75 + 6.50i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-4.53 + 7.85i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.98 + 6.90i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.14127136430325395478338721729, −8.780210904012332709238677659699, −8.022570012109731511506217040863, −7.57351907355223970716549064128, −6.43541941143878906997211473662, −5.95449143721090686740368407306, −5.10348980165114368427630979712, −4.19326832008263646997867208591, −3.32811648214157249347660564913, −1.67877012269408684001893183902,
0.74056972453790423242065736501, 2.23788298637031179939629658456, 2.96003217703926217581328118601, 3.93061654062352131403673293181, 4.94203718574208915285190088960, 5.43362389473191061171253079556, 6.90770086190929098580227621257, 7.53143072058734623861675774655, 8.744406650918856873207685790538, 9.512913621031152551260563369492