| L(s) = 1 | + (1.96 − 4.74i)2-s + (−7.55 − 5.04i)3-s + (−7.37 − 7.37i)4-s + (12.5 − 2.49i)5-s + (−38.8 + 25.9i)6-s + (32.9 + 6.56i)7-s + (26.4 − 10.9i)8-s + (0.600 + 1.45i)9-s + (12.8 − 64.4i)10-s + (71.8 + 107. i)11-s + (18.4 + 92.9i)12-s + (−186. + 186. i)13-s + (96.0 − 143. i)14-s + (−107. − 44.4i)15-s − 314. i·16-s + (−122. + 261. i)17-s + ⋯ |
| L(s) = 1 | + (0.491 − 1.18i)2-s + (−0.839 − 0.560i)3-s + (−0.461 − 0.461i)4-s + (0.501 − 0.0996i)5-s + (−1.07 + 0.720i)6-s + (0.673 + 0.133i)7-s + (0.413 − 0.171i)8-s + (0.00741 + 0.0179i)9-s + (0.128 − 0.644i)10-s + (0.593 + 0.888i)11-s + (0.128 + 0.645i)12-s + (−1.10 + 1.10i)13-s + (0.490 − 0.733i)14-s + (−0.476 − 0.197i)15-s − 1.22i·16-s + (−0.422 + 0.906i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.280 + 0.959i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.280 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.810603 - 1.08110i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.810603 - 1.08110i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (122. - 261. i)T \) |
| good | 2 | \( 1 + (-1.96 + 4.74i)T + (-11.3 - 11.3i)T^{2} \) |
| 3 | \( 1 + (7.55 + 5.04i)T + (30.9 + 74.8i)T^{2} \) |
| 5 | \( 1 + (-12.5 + 2.49i)T + (577. - 239. i)T^{2} \) |
| 7 | \( 1 + (-32.9 - 6.56i)T + (2.21e3 + 918. i)T^{2} \) |
| 11 | \( 1 + (-71.8 - 107. i)T + (-5.60e3 + 1.35e4i)T^{2} \) |
| 13 | \( 1 + (186. - 186. i)T - 2.85e4iT^{2} \) |
| 19 | \( 1 + (-186. + 449. i)T + (-9.21e4 - 9.21e4i)T^{2} \) |
| 23 | \( 1 + (-123. + 82.7i)T + (1.07e5 - 2.58e5i)T^{2} \) |
| 29 | \( 1 + (-94.2 - 473. i)T + (-6.53e5 + 2.70e5i)T^{2} \) |
| 31 | \( 1 + (-425. + 636. i)T + (-3.53e5 - 8.53e5i)T^{2} \) |
| 37 | \( 1 + (-1.23e3 - 827. i)T + (7.17e5 + 1.73e6i)T^{2} \) |
| 41 | \( 1 + (1.19e3 + 237. i)T + (2.61e6 + 1.08e6i)T^{2} \) |
| 43 | \( 1 + (675. + 1.63e3i)T + (-2.41e6 + 2.41e6i)T^{2} \) |
| 47 | \( 1 + (2.15e3 - 2.15e3i)T - 4.87e6iT^{2} \) |
| 53 | \( 1 + (-1.17e3 + 2.82e3i)T + (-5.57e6 - 5.57e6i)T^{2} \) |
| 59 | \( 1 + (879. - 364. i)T + (8.56e6 - 8.56e6i)T^{2} \) |
| 61 | \( 1 + (-330. + 1.66e3i)T + (-1.27e7 - 5.29e6i)T^{2} \) |
| 67 | \( 1 + 389. iT - 2.01e7T^{2} \) |
| 71 | \( 1 + (-4.47e3 - 2.99e3i)T + (9.72e6 + 2.34e7i)T^{2} \) |
| 73 | \( 1 + (9.65e3 - 1.92e3i)T + (2.62e7 - 1.08e7i)T^{2} \) |
| 79 | \( 1 + (3.95e3 + 5.91e3i)T + (-1.49e7 + 3.59e7i)T^{2} \) |
| 83 | \( 1 + (-7.61e3 - 3.15e3i)T + (3.35e7 + 3.35e7i)T^{2} \) |
| 89 | \( 1 + (-8.11e3 - 8.11e3i)T + 6.27e7iT^{2} \) |
| 97 | \( 1 + (737. + 3.70e3i)T + (-8.17e7 + 3.38e7i)T^{2} \) |
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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
−17.75038341736753677957016763595, −16.99351542715466616870715633459, −14.71477720926018388072200118489, −13.21659516903694316723327337719, −12.03265624224874144700097706796, −11.34731926485613125284932515131, −9.613305494626273961807800574743, −6.90258740200466736902139264064, −4.74514576704170082872540970170, −1.80612764267945605898502251412,
4.92896160927056932388419020953, 5.99123377270711377112044162352, 7.85580821718464337832108395932, 10.18530665963670277262222119281, 11.57846584747724135696241154268, 13.68359788365130489627230613934, 14.68666108534769263243261038165, 16.03699401295191040434795656526, 16.94536182088963026294927586171, 17.81395308605128577374255276763
