| L(s) = 1 | + (19.5 + 11.3i)2-s + (1.63 − 242. i)3-s + (255. + 443. i)4-s + (−1.94e3 + 1.12e3i)5-s + (2.78e3 − 4.74e3i)6-s + (1.48e4 − 2.57e4i)7-s + 1.15e4i·8-s + (−5.90e4 − 796. i)9-s − 5.08e4·10-s + (−1.39e5 − 8.06e4i)11-s + (1.08e5 − 6.14e4i)12-s + (−2.49e5 − 4.31e5i)13-s + (5.82e5 − 3.36e5i)14-s + (2.69e5 + 4.74e5i)15-s + (−1.31e5 + 2.27e5i)16-s − 1.34e6i·17-s + ⋯ |
| L(s) = 1 | + (0.612 + 0.353i)2-s + (0.00674 − 0.999i)3-s + (0.249 + 0.433i)4-s + (−0.623 + 0.359i)5-s + (0.357 − 0.609i)6-s + (0.884 − 1.53i)7-s + 0.353i·8-s + (−0.999 − 0.0134i)9-s − 0.508·10-s + (−0.867 − 0.500i)11-s + (0.434 − 0.247i)12-s + (−0.671 − 1.16i)13-s + (1.08 − 0.625i)14-s + (0.355 + 0.625i)15-s + (−0.125 + 0.216i)16-s − 0.947i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.186 + 0.982i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.186 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(1.20365 - 1.45427i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.20365 - 1.45427i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-19.5 - 11.3i)T \) |
| 3 | \( 1 + (-1.63 + 242. i)T \) |
| good | 5 | \( 1 + (1.94e3 - 1.12e3i)T + (4.88e6 - 8.45e6i)T^{2} \) |
| 7 | \( 1 + (-1.48e4 + 2.57e4i)T + (-1.41e8 - 2.44e8i)T^{2} \) |
| 11 | \( 1 + (1.39e5 + 8.06e4i)T + (1.29e10 + 2.24e10i)T^{2} \) |
| 13 | \( 1 + (2.49e5 + 4.31e5i)T + (-6.89e10 + 1.19e11i)T^{2} \) |
| 17 | \( 1 + 1.34e6iT - 2.01e12T^{2} \) |
| 19 | \( 1 - 4.27e6T + 6.13e12T^{2} \) |
| 23 | \( 1 + (3.03e6 - 1.75e6i)T + (2.07e13 - 3.58e13i)T^{2} \) |
| 29 | \( 1 + (-1.44e7 - 8.32e6i)T + (2.10e14 + 3.64e14i)T^{2} \) |
| 31 | \( 1 + (-5.30e6 - 9.19e6i)T + (-4.09e14 + 7.09e14i)T^{2} \) |
| 37 | \( 1 + 3.61e6T + 4.80e15T^{2} \) |
| 41 | \( 1 + (-1.48e7 + 8.59e6i)T + (6.71e15 - 1.16e16i)T^{2} \) |
| 43 | \( 1 + (-7.92e7 + 1.37e8i)T + (-1.08e16 - 1.87e16i)T^{2} \) |
| 47 | \( 1 + (-1.61e8 - 9.32e7i)T + (2.62e16 + 4.55e16i)T^{2} \) |
| 53 | \( 1 - 6.84e8iT - 1.74e17T^{2} \) |
| 59 | \( 1 + (-1.08e8 + 6.28e7i)T + (2.55e17 - 4.42e17i)T^{2} \) |
| 61 | \( 1 + (-3.64e8 + 6.31e8i)T + (-3.56e17 - 6.17e17i)T^{2} \) |
| 67 | \( 1 + (1.98e8 + 3.43e8i)T + (-9.11e17 + 1.57e18i)T^{2} \) |
| 71 | \( 1 + 2.21e9iT - 3.25e18T^{2} \) |
| 73 | \( 1 - 6.41e8T + 4.29e18T^{2} \) |
| 79 | \( 1 + (-1.46e9 + 2.54e9i)T + (-4.73e18 - 8.19e18i)T^{2} \) |
| 83 | \( 1 + (1.18e9 + 6.83e8i)T + (7.75e18 + 1.34e19i)T^{2} \) |
| 89 | \( 1 + 6.25e9iT - 3.11e19T^{2} \) |
| 97 | \( 1 + (3.15e9 - 5.46e9i)T + (-3.68e19 - 6.38e19i)T^{2} \) |
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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
−15.78840535997494407628359349298, −14.20102610660413620058260185962, −13.53259424899115549171101395133, −11.96625702267590143676207846735, −10.76649509259247461489694515507, −7.71361143278423195361047851821, −7.43018793220643174302704224712, −5.22233606885039852913381922554, −3.13776557197626920441650710966, −0.69650368064368702835346536604,
2.40006786794799974464692664715, 4.40666615693135088851119809481, 5.47234564483050704645684343990, 8.254974194779930402153405582165, 9.749557158527949614474353361951, 11.49370859838403023278828993746, 12.19167101216001842733549253895, 14.29742677141776892542059304385, 15.30082241642498593230167990206, 16.11808387804499566840224322251
