| L(s) = 1 | + (0.218 − 0.673i)2-s + (7.28 − 5.29i)3-s + (25.4 + 18.5i)4-s + (−21.0 − 64.6i)5-s + (−1.96 − 6.05i)6-s + (98.3 + 71.4i)7-s + (36.3 − 26.4i)8-s + (25.0 − 77.0i)9-s − 48.1·10-s + (251. − 312. i)11-s + 283.·12-s + (235. − 725. i)13-s + (69.6 − 50.5i)14-s + (−495. − 359. i)15-s + (301. + 928. i)16-s + (468. + 1.44e3i)17-s + ⋯ |
| L(s) = 1 | + (0.0386 − 0.118i)2-s + (0.467 − 0.339i)3-s + (0.796 + 0.578i)4-s + (−0.375 − 1.15i)5-s + (−0.0223 − 0.0686i)6-s + (0.758 + 0.551i)7-s + (0.200 − 0.145i)8-s + (0.103 − 0.317i)9-s − 0.152·10-s + (0.626 − 0.779i)11-s + 0.568·12-s + (0.386 − 1.19i)13-s + (0.0949 − 0.0689i)14-s + (−0.568 − 0.412i)15-s + (0.294 + 0.906i)16-s + (0.392 + 1.20i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.807 + 0.589i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.807 + 0.589i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.02322 - 0.660040i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.02322 - 0.660040i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-7.28 + 5.29i)T \) |
| 11 | \( 1 + (-251. + 312. i)T \) |
| good | 2 | \( 1 + (-0.218 + 0.673i)T + (-25.8 - 18.8i)T^{2} \) |
| 5 | \( 1 + (21.0 + 64.6i)T + (-2.52e3 + 1.83e3i)T^{2} \) |
| 7 | \( 1 + (-98.3 - 71.4i)T + (5.19e3 + 1.59e4i)T^{2} \) |
| 13 | \( 1 + (-235. + 725. i)T + (-3.00e5 - 2.18e5i)T^{2} \) |
| 17 | \( 1 + (-468. - 1.44e3i)T + (-1.14e6 + 8.34e5i)T^{2} \) |
| 19 | \( 1 + (2.01e3 - 1.46e3i)T + (7.65e5 - 2.35e6i)T^{2} \) |
| 23 | \( 1 + 3.94e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + (-178. - 129. i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 31 | \( 1 + (2.88e3 - 8.87e3i)T + (-2.31e7 - 1.68e7i)T^{2} \) |
| 37 | \( 1 + (1.61e3 + 1.17e3i)T + (2.14e7 + 6.59e7i)T^{2} \) |
| 41 | \( 1 + (-5.16e3 + 3.75e3i)T + (3.58e7 - 1.10e8i)T^{2} \) |
| 43 | \( 1 - 8.05e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (8.20e3 - 5.95e3i)T + (7.08e7 - 2.18e8i)T^{2} \) |
| 53 | \( 1 + (1.92e3 - 5.93e3i)T + (-3.38e8 - 2.45e8i)T^{2} \) |
| 59 | \( 1 + (1.93e4 + 1.40e4i)T + (2.20e8 + 6.79e8i)T^{2} \) |
| 61 | \( 1 + (1.96e3 + 6.04e3i)T + (-6.83e8 + 4.96e8i)T^{2} \) |
| 67 | \( 1 - 4.07e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (-9.41e3 - 2.89e4i)T + (-1.45e9 + 1.06e9i)T^{2} \) |
| 73 | \( 1 + (-5.43e3 - 3.94e3i)T + (6.40e8 + 1.97e9i)T^{2} \) |
| 79 | \( 1 + (-2.83e4 + 8.73e4i)T + (-2.48e9 - 1.80e9i)T^{2} \) |
| 83 | \( 1 + (-1.87e4 - 5.76e4i)T + (-3.18e9 + 2.31e9i)T^{2} \) |
| 89 | \( 1 + 9.89e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (2.67e4 - 8.22e4i)T + (-6.94e9 - 5.04e9i)T^{2} \) |
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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
−15.66592640389755408691830726945, −14.42715944099141719438370980058, −12.67379322369557635328564456572, −12.24637315001326605729803122210, −10.76346535668876919979152727953, −8.410355270068734704194948803330, −8.179405747971109528306796777038, −5.97409472854001987279295522402, −3.76071737033392342208358943887, −1.61197726437267692237042275420,
2.16944926260302948889605076544, 4.27057878177090954229442762361, 6.59815732523553191684393711893, 7.58062450972734741911201195220, 9.613937475336028034019618206366, 10.91410709525440352388847559232, 11.63140444326948128864811205165, 14.02409286809718754007503489016, 14.61167735118101233883105456108, 15.51898826404043146854014562639
