| L(s) = 1 | + (0.718 + 0.829i)2-s + (0.397 − 2.76i)4-s + (4.86 + 2.21i)5-s + (2.07 − 7.07i)7-s + (6.27 − 4.03i)8-s + (1.65 + 5.62i)10-s + (−10.0 − 8.71i)11-s + (−17.3 + 5.08i)13-s + (7.36 − 3.36i)14-s + (−2.87 − 0.844i)16-s + (30.2 − 4.35i)17-s + (29.7 + 4.28i)19-s + (8.07 − 12.5i)20-s − 14.6i·22-s + (15.5 + 16.9i)23-s + ⋯ |
| L(s) = 1 | + (0.359 + 0.414i)2-s + (0.0994 − 0.691i)4-s + (0.972 + 0.443i)5-s + (0.296 − 1.01i)7-s + (0.784 − 0.503i)8-s + (0.165 + 0.562i)10-s + (−0.914 − 0.792i)11-s + (−1.33 + 0.390i)13-s + (0.526 − 0.240i)14-s + (−0.179 − 0.0527i)16-s + (1.78 − 0.256i)17-s + (1.56 + 0.225i)19-s + (0.403 − 0.628i)20-s − 0.663i·22-s + (0.675 + 0.736i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.930 + 0.366i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.930 + 0.366i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.15877 - 0.409709i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.15877 - 0.409709i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + (-15.5 - 16.9i)T \) |
| good | 2 | \( 1 + (-0.718 - 0.829i)T + (-0.569 + 3.95i)T^{2} \) |
| 5 | \( 1 + (-4.86 - 2.21i)T + (16.3 + 18.8i)T^{2} \) |
| 7 | \( 1 + (-2.07 + 7.07i)T + (-41.2 - 26.4i)T^{2} \) |
| 11 | \( 1 + (10.0 + 8.71i)T + (17.2 + 119. i)T^{2} \) |
| 13 | \( 1 + (17.3 - 5.08i)T + (142. - 91.3i)T^{2} \) |
| 17 | \( 1 + (-30.2 + 4.35i)T + (277. - 81.4i)T^{2} \) |
| 19 | \( 1 + (-29.7 - 4.28i)T + (346. + 101. i)T^{2} \) |
| 29 | \( 1 + (-4.53 - 31.5i)T + (-806. + 236. i)T^{2} \) |
| 31 | \( 1 + (11.7 - 7.54i)T + (399. - 874. i)T^{2} \) |
| 37 | \( 1 + (60.2 - 27.5i)T + (896. - 1.03e3i)T^{2} \) |
| 41 | \( 1 + (-3.19 + 6.99i)T + (-1.10e3 - 1.27e3i)T^{2} \) |
| 43 | \( 1 + (-19.3 + 30.1i)T + (-768. - 1.68e3i)T^{2} \) |
| 47 | \( 1 + 39.0T + 2.20e3T^{2} \) |
| 53 | \( 1 + (27.4 - 93.6i)T + (-2.36e3 - 1.51e3i)T^{2} \) |
| 59 | \( 1 + (12.8 - 3.77i)T + (2.92e3 - 1.88e3i)T^{2} \) |
| 61 | \( 1 + (3.69 + 5.75i)T + (-1.54e3 + 3.38e3i)T^{2} \) |
| 67 | \( 1 + (1.90 - 1.64i)T + (638. - 4.44e3i)T^{2} \) |
| 71 | \( 1 + (-65.1 - 75.1i)T + (-717. + 4.98e3i)T^{2} \) |
| 73 | \( 1 + (-8.23 + 57.2i)T + (-5.11e3 - 1.50e3i)T^{2} \) |
| 79 | \( 1 + (14.5 + 49.3i)T + (-5.25e3 + 3.37e3i)T^{2} \) |
| 83 | \( 1 + (51.3 - 23.4i)T + (4.51e3 - 5.20e3i)T^{2} \) |
| 89 | \( 1 + (39.9 - 62.2i)T + (-3.29e3 - 7.20e3i)T^{2} \) |
| 97 | \( 1 + (-1.71 - 0.785i)T + (6.16e3 + 7.11e3i)T^{2} \) |
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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
−12.17970566991557834942289883252, −10.83644656317140526406987396536, −10.17449100379761661640973756403, −9.526478664415663626755961181466, −7.61127878128017497681272481764, −7.04119589820743410825152975303, −5.57507806379930440538875611817, −5.10462533694075138302404442935, −3.16391589859926935067085936897, −1.30212986834237041892421077904,
2.02803337340285547346051697216, 3.02717913032507162598094867957, 5.02492218109860375762636575486, 5.43170534479044429908037663620, 7.34791253917862265010220156201, 8.130689753365384869711903431146, 9.487925470335707342167289870282, 10.14220672168427885416182415982, 11.58862100683813766735760753959, 12.44787441045099297952548729536
