L(s) = 1 | + 26.2·2-s + (17.3 − 242. i)3-s − 333.·4-s + (3.04e3 + 714. i)5-s + (456. − 6.36e3i)6-s − 2.21e4i·7-s − 3.56e4·8-s + (−5.84e4 − 8.41e3i)9-s + (7.99e4 + 1.87e4i)10-s − 2.12e5i·11-s + (−5.79e3 + 8.08e4i)12-s + 1.46e5i·13-s − 5.81e5i·14-s + (2.26e5 − 7.24e5i)15-s − 5.95e5·16-s − 1.94e5·17-s + ⋯ |
L(s) = 1 | + 0.821·2-s + (0.0714 − 0.997i)3-s − 0.325·4-s + (0.973 + 0.228i)5-s + (0.0586 − 0.819i)6-s − 1.31i·7-s − 1.08·8-s + (−0.989 − 0.142i)9-s + (0.799 + 0.187i)10-s − 1.32i·11-s + (−0.0232 + 0.324i)12-s + 0.395i·13-s − 1.08i·14-s + (0.297 − 0.954i)15-s − 0.568·16-s − 0.137·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.297 + 0.954i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.297 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{11}{2})\) |
\(\approx\) |
\(1.37515 - 1.86937i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.37515 - 1.86937i\) |
\(L(6)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-17.3 + 242. i)T \) |
| 5 | \( 1 + (-3.04e3 - 714. i)T \) |
good | 2 | \( 1 - 26.2T + 1.02e3T^{2} \) |
| 7 | \( 1 + 2.21e4iT - 2.82e8T^{2} \) |
| 11 | \( 1 + 2.12e5iT - 2.59e10T^{2} \) |
| 13 | \( 1 - 1.46e5iT - 1.37e11T^{2} \) |
| 17 | \( 1 + 1.94e5T + 2.01e12T^{2} \) |
| 19 | \( 1 - 2.72e6T + 6.13e12T^{2} \) |
| 23 | \( 1 - 8.45e6T + 4.14e13T^{2} \) |
| 29 | \( 1 - 2.69e7iT - 4.20e14T^{2} \) |
| 31 | \( 1 + 1.28e7T + 8.19e14T^{2} \) |
| 37 | \( 1 + 3.69e7iT - 4.80e15T^{2} \) |
| 41 | \( 1 + 1.87e8iT - 1.34e16T^{2} \) |
| 43 | \( 1 - 1.57e8iT - 2.16e16T^{2} \) |
| 47 | \( 1 - 1.71e8T + 5.25e16T^{2} \) |
| 53 | \( 1 - 9.61e7T + 1.74e17T^{2} \) |
| 59 | \( 1 + 6.71e8iT - 5.11e17T^{2} \) |
| 61 | \( 1 + 2.00e8T + 7.13e17T^{2} \) |
| 67 | \( 1 + 1.13e8iT - 1.82e18T^{2} \) |
| 71 | \( 1 + 1.53e9iT - 3.25e18T^{2} \) |
| 73 | \( 1 - 3.09e9iT - 4.29e18T^{2} \) |
| 79 | \( 1 + 4.13e9T + 9.46e18T^{2} \) |
| 83 | \( 1 - 7.78e8T + 1.55e19T^{2} \) |
| 89 | \( 1 - 7.12e9iT - 3.11e19T^{2} \) |
| 97 | \( 1 + 7.15e8iT - 7.37e19T^{2} \) |
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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
−16.75800308923832417686526641404, −14.23943534217140045877678738344, −13.81342704915348430056117833815, −12.87191416068920556016944705563, −11.03681003139161119318305411805, −9.015115864441013405549746355637, −6.96910179857268307165064853648, −5.51449993161777482051671848524, −3.24296654680996543511205882262, −0.917631407688182055891838976609,
2.72305108897558574650251994252, 4.79732063215564711774797145033, 5.73085338439392935925970570702, 8.960974624539586320108392492360, 9.815640932834279084360741142375, 11.97497054338034869835044328244, 13.28237795134340759227762941291, 14.72032623048625490455689381128, 15.46014760146925835070352927692, 17.29196128086709384906406844271