L(s) = 1 | − 7.06i·2-s + (−15.4 + 1.75i)3-s − 17.9·4-s − 82.3·5-s + (12.3 + 109. i)6-s + 96.3i·7-s − 99.5i·8-s + (236. − 54.2i)9-s + 581. i·10-s + 87.8·11-s + (277. − 31.3i)12-s + 303.·13-s + 680.·14-s + (1.27e3 − 144. i)15-s − 1.27e3·16-s + 460.·17-s + ⋯ |
L(s) = 1 | − 1.24i·2-s + (−0.993 + 0.112i)3-s − 0.559·4-s − 1.47·5-s + (0.140 + 1.24i)6-s + 0.742i·7-s − 0.550i·8-s + (0.974 − 0.223i)9-s + 1.83i·10-s + 0.218·11-s + (0.555 − 0.0628i)12-s + 0.498·13-s + 0.927·14-s + (1.46 − 0.165i)15-s − 1.24·16-s + 0.386·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.163i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.986 + 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.766505 - 0.0630573i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.766505 - 0.0630573i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (15.4 - 1.75i)T \) |
| 23 | \( 1 + (-2.44e3 - 693. i)T \) |
good | 2 | \( 1 + 7.06iT - 32T^{2} \) |
| 5 | \( 1 + 82.3T + 3.12e3T^{2} \) |
| 7 | \( 1 - 96.3iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 87.8T + 1.61e5T^{2} \) |
| 13 | \( 1 - 303.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 460.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.56e3iT - 2.47e6T^{2} \) |
| 29 | \( 1 + 4.09e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 1.86e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.18e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.21e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 2.24e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 2.74e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 1.15e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.99e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 2.44e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 4.17e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 3.38e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 - 6.36e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.16e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + 7.23e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 9.26e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 2.84e4iT - 8.58e9T^{2} \) |
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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
−13.06015873042124710106510130590, −12.04870144417087756548266707678, −11.66636074599789773204781586106, −10.79762330055780257097447993841, −9.557676061084146956620880923444, −7.919000334857987590046942515281, −6.35447956792023599308444086746, −4.54513804222226550276173583497, −3.32750665400719665752396062895, −1.11606198376620570384242195417,
0.49467219593005663050569877142, 4.04356844983768289509769111858, 5.30658064244650702727520461899, 6.90519136940703332598744719200, 7.34687852976921279830052304693, 8.676934170213706982991406034351, 10.76063356169045018948808791918, 11.44375609787975141083985835602, 12.62922499128363811973839536894, 14.04199101397497883880544775623