L(s) = 1 | + (−0.483 − 1.32i)2-s + (−1.53 + 1.28i)4-s + (−0.891 + 0.157i)5-s + (−5.50 − 4.61i)7-s + (2.44 + 1.41i)8-s + (0.639 + 1.10i)10-s + (−7.28 − 1.28i)11-s + (−18.3 − 6.68i)13-s + (−3.47 + 9.54i)14-s + (0.694 − 3.93i)16-s + (−12.7 + 7.37i)17-s + (6.58 − 11.4i)19-s + (1.16 − 1.38i)20-s + (1.81 + 10.2i)22-s + (15.0 + 17.9i)23-s + ⋯ |
L(s) = 1 | + (−0.241 − 0.664i)2-s + (−0.383 + 0.321i)4-s + (−0.178 + 0.0314i)5-s + (−0.785 − 0.659i)7-s + (0.306 + 0.176i)8-s + (0.0639 + 0.110i)10-s + (−0.662 − 0.116i)11-s + (−1.41 − 0.514i)13-s + (−0.248 + 0.681i)14-s + (0.0434 − 0.246i)16-s + (−0.751 + 0.434i)17-s + (0.346 − 0.600i)19-s + (0.0581 − 0.0693i)20-s + (0.0825 + 0.468i)22-s + (0.655 + 0.780i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 - 0.220i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.975 - 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0380807 + 0.340417i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0380807 + 0.340417i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.483 + 1.32i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (0.891 - 0.157i)T + (23.4 - 8.55i)T^{2} \) |
| 7 | \( 1 + (5.50 + 4.61i)T + (8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (7.28 + 1.28i)T + (113. + 41.3i)T^{2} \) |
| 13 | \( 1 + (18.3 + 6.68i)T + (129. + 108. i)T^{2} \) |
| 17 | \( 1 + (12.7 - 7.37i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-6.58 + 11.4i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-15.0 - 17.9i)T + (-91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (13.5 + 37.1i)T + (-644. + 540. i)T^{2} \) |
| 31 | \( 1 + (9.20 - 7.72i)T + (166. - 946. i)T^{2} \) |
| 37 | \( 1 + (-33.0 - 57.1i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-26.1 + 71.8i)T + (-1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (-9.66 + 54.8i)T + (-1.73e3 - 632. i)T^{2} \) |
| 47 | \( 1 + (-0.204 + 0.243i)T + (-383. - 2.17e3i)T^{2} \) |
| 53 | \( 1 - 0.264iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-62.3 + 10.9i)T + (3.27e3 - 1.19e3i)T^{2} \) |
| 61 | \( 1 + (42.4 + 35.6i)T + (646. + 3.66e3i)T^{2} \) |
| 67 | \( 1 + (-58.6 - 21.3i)T + (3.43e3 + 2.88e3i)T^{2} \) |
| 71 | \( 1 + (57.0 - 32.9i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-22.7 + 39.3i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-22.8 + 8.33i)T + (4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (-28.1 - 77.2i)T + (-5.27e3 + 4.42e3i)T^{2} \) |
| 89 | \( 1 + (115. + 66.6i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (11.9 - 67.8i)T + (-8.84e3 - 3.21e3i)T^{2} \) |
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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
−12.05875006829457197482161572456, −11.01908332679347376692899275122, −10.07745275607811273424667083015, −9.348725497156665066546107234338, −7.902602916323703633272458073959, −7.02118956327086471227884855392, −5.32449291816301313866117845711, −3.89053759238096019697475250168, −2.54744758824916884802807510367, −0.21594298905324959195187762523,
2.61396337296714239902275517521, 4.53735688661115520986311780372, 5.73836484517496927193239222345, 6.92108247149089276914912587460, 7.86489685725753688195289042233, 9.191572770403969283821241236006, 9.772561928516672966040204397756, 11.11989238789413097749022218377, 12.39175757303657712144395042243, 13.05488352538648934521581236531