| L(s) = 1 | + (1.09 − 0.900i)2-s + (−0.540 − 0.841i)3-s + (0.377 − 1.96i)4-s + (2.39 − 1.09i)5-s + (−1.34 − 0.430i)6-s + (−2.35 + 0.692i)7-s + (−1.35 − 2.48i)8-s + (−0.415 + 0.909i)9-s + (1.62 − 3.35i)10-s + (−0.501 + 0.434i)11-s + (−1.85 + 0.744i)12-s + (1.53 − 5.22i)13-s + (−1.94 + 2.87i)14-s + (−2.21 − 1.42i)15-s + (−3.71 − 1.48i)16-s + (0.240 − 1.67i)17-s + ⋯ |
| L(s) = 1 | + (0.771 − 0.636i)2-s + (−0.312 − 0.485i)3-s + (0.188 − 0.981i)4-s + (1.07 − 0.490i)5-s + (−0.549 − 0.175i)6-s + (−0.890 + 0.261i)7-s + (−0.479 − 0.877i)8-s + (−0.138 + 0.303i)9-s + (0.515 − 1.06i)10-s + (−0.151 + 0.131i)11-s + (−0.535 + 0.214i)12-s + (0.425 − 1.44i)13-s + (−0.520 + 0.769i)14-s + (−0.572 − 0.368i)15-s + (−0.928 − 0.370i)16-s + (0.0583 − 0.405i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.690 + 0.723i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.690 + 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.807519 - 1.88781i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.807519 - 1.88781i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.09 + 0.900i)T \) |
| 3 | \( 1 + (0.540 + 0.841i)T \) |
| 23 | \( 1 + (1.19 - 4.64i)T \) |
| good | 5 | \( 1 + (-2.39 + 1.09i)T + (3.27 - 3.77i)T^{2} \) |
| 7 | \( 1 + (2.35 - 0.692i)T + (5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (0.501 - 0.434i)T + (1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-1.53 + 5.22i)T + (-10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.240 + 1.67i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (-1.89 + 0.272i)T + (18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-1.46 - 0.210i)T + (27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (-0.835 - 0.536i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (-4.40 - 2.01i)T + (24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (3.22 + 7.06i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-2.48 - 3.86i)T + (-17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 - 7.28T + 47T^{2} \) |
| 53 | \( 1 + (-1.74 - 5.95i)T + (-44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (-0.364 + 1.24i)T + (-49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (-3.42 + 5.32i)T + (-25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (1.27 + 1.10i)T + (9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (-8.75 + 10.1i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (-1.92 - 13.4i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (-1.12 - 0.330i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (3.16 + 1.44i)T + (54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (-6.93 + 4.45i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (-3.48 - 7.62i)T + (-63.5 + 73.3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.44843971798811872870012415415, −9.801869928493152740997320762963, −9.041480633395850956707084397800, −7.60322538008700357197161286985, −6.35739089267259760743812928678, −5.71301491341509162980598771731, −5.08643636909263273542606076499, −3.43950557365594686538505444645, −2.40467232430837173760873134792, −0.995610215433738040353701558216,
2.36536953381993255785793253192, 3.59529908340190020140434553092, 4.55931665438807586919944861553, 5.82611164978462349777877184886, 6.36014212199167029257631456245, 7.04553665790250925426071484745, 8.481745941267611449964509766838, 9.435498577226673720042754947989, 10.18614308387011465822989134987, 11.14535738798907525720695376036
