L(s) = 1 | + (0.765 + 5.60i)2-s + 17.3i·3-s + (−30.8 + 8.57i)4-s + 25i·5-s + (−97.0 + 13.2i)6-s − 9.19·7-s + (−71.6 − 166. i)8-s − 56.8·9-s + (−140. + 19.1i)10-s + 160. i·11-s + (−148. − 533. i)12-s − 368. i·13-s + (−7.03 − 51.5i)14-s − 432.·15-s + (876. − 528. i)16-s − 1.26e3·17-s + ⋯ |
L(s) = 1 | + (0.135 + 0.990i)2-s + 1.11i·3-s + (−0.963 + 0.268i)4-s + 0.447i·5-s + (−1.10 + 0.150i)6-s − 0.0708·7-s + (−0.395 − 0.918i)8-s − 0.233·9-s + (−0.443 + 0.0604i)10-s + 0.399i·11-s + (−0.297 − 1.07i)12-s − 0.604i·13-s + (−0.00958 − 0.0702i)14-s − 0.496·15-s + (0.856 − 0.516i)16-s − 1.05·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.918 + 0.395i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.918 + 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.251481 - 1.21852i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.251481 - 1.21852i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.765 - 5.60i)T \) |
| 5 | \( 1 - 25iT \) |
good | 3 | \( 1 - 17.3iT - 243T^{2} \) |
| 7 | \( 1 + 9.19T + 1.68e4T^{2} \) |
| 11 | \( 1 - 160. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + 368. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 1.26e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.48e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 422.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.66e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 9.38e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.56e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 5.94e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.06e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 9.24e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 8.97e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 2.74e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 5.05e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 5.96e3iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 6.72e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 8.57e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.65e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.02e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 1.13e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.38e5T + 8.58e9T^{2} \) |
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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
−15.62491546472039742523793433575, −15.01040813284074318794588460803, −13.87257186900404214230338944171, −12.44368354577939447796673037125, −10.54459785363828731243801764390, −9.626090344103047718901483784114, −8.188494609838226486908530402234, −6.61428942665751244713049649166, −5.03310369004182162750710361338, −3.68257727971470362261367562087,
0.69280420922858683866645964031, 2.32198114007091358806659784116, 4.53643788785416897609963132828, 6.48948822270616495069229571163, 8.251764503294293200993432746839, 9.490705227205113866218250110034, 11.18023072865742854293285037914, 12.12388599050166477140121135457, 13.28226565804638980831587232350, 13.74034248867147023610426526881