L(s) = 1 | + (3.53 − 4.41i)2-s + (6.36 + 6.36i)3-s + (−6.97 − 31.2i)4-s + (37.1 − 37.1i)5-s + (50.6 − 5.58i)6-s − 5.83i·7-s + (−162. − 79.6i)8-s + 81i·9-s + (−32.6 − 295. i)10-s + (461. − 461. i)11-s + (154. − 243. i)12-s + (−43.8 − 43.8i)13-s + (−25.7 − 20.6i)14-s + 473.·15-s + (−926. + 435. i)16-s − 365.·17-s + ⋯ |
L(s) = 1 | + (0.625 − 0.780i)2-s + (0.408 + 0.408i)3-s + (−0.218 − 0.975i)4-s + (0.665 − 0.665i)5-s + (0.573 − 0.0633i)6-s − 0.0449i·7-s + (−0.897 − 0.440i)8-s + 0.333i·9-s + (−0.103 − 0.935i)10-s + (1.14 − 1.14i)11-s + (0.309 − 0.487i)12-s + (−0.0720 − 0.0720i)13-s + (−0.0351 − 0.0281i)14-s + 0.543·15-s + (−0.904 + 0.425i)16-s − 0.307·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0469 + 0.998i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.0469 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.96118 - 1.87111i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.96118 - 1.87111i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-3.53 + 4.41i)T \) |
| 3 | \( 1 + (-6.36 - 6.36i)T \) |
good | 5 | \( 1 + (-37.1 + 37.1i)T - 3.12e3iT^{2} \) |
| 7 | \( 1 + 5.83iT - 1.68e4T^{2} \) |
| 11 | \( 1 + (-461. + 461. i)T - 1.61e5iT^{2} \) |
| 13 | \( 1 + (43.8 + 43.8i)T + 3.71e5iT^{2} \) |
| 17 | \( 1 + 365.T + 1.41e6T^{2} \) |
| 19 | \( 1 + (322. + 322. i)T + 2.47e6iT^{2} \) |
| 23 | \( 1 - 2.75e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + (-4.46e3 - 4.46e3i)T + 2.05e7iT^{2} \) |
| 31 | \( 1 - 4.84e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + (5.00e3 - 5.00e3i)T - 6.93e7iT^{2} \) |
| 41 | \( 1 - 1.40e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 + (-9.38e3 + 9.38e3i)T - 1.47e8iT^{2} \) |
| 47 | \( 1 + 2.20e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + (2.01e4 - 2.01e4i)T - 4.18e8iT^{2} \) |
| 59 | \( 1 + (-6.59e3 + 6.59e3i)T - 7.14e8iT^{2} \) |
| 61 | \( 1 + (-4.88e3 - 4.88e3i)T + 8.44e8iT^{2} \) |
| 67 | \( 1 + (2.74e4 + 2.74e4i)T + 1.35e9iT^{2} \) |
| 71 | \( 1 + 7.93e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + 7.79e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 7.41e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + (-6.69e4 - 6.69e4i)T + 3.93e9iT^{2} \) |
| 89 | \( 1 - 6.84e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + 1.86e4T + 8.58e9T^{2} \) |
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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
−13.99728012090148105015229757837, −13.44384017999981707928313536324, −12.07196301212298055348690230943, −10.89465176204694077276012020193, −9.565334899309207516615202561437, −8.741268119433035266342609771643, −6.22049875173849687966956419023, −4.82764349528597900264463609470, −3.29236060147463270464356988296, −1.33669322401149950670709057969,
2.41148640191782215751661714411, 4.30035150056943724905528949533, 6.26510355684618144977079900555, 7.04267916980857968967462634340, 8.576176755100726479667974529438, 9.935715168288201153052900955626, 11.85659746475133598575189059059, 12.85980457060154611598696893555, 14.16999971636869553667127226468, 14.55570409653553861147251491708