| L(s) = 1 | + (−0.461 − 1.33i)2-s + (−1.07 − 1.15i)3-s + (−1.57 + 1.23i)4-s + (−1.26 − 2.78i)5-s + (−1.04 + 1.97i)6-s + (2.47 + 0.933i)7-s + (2.37 + 1.53i)8-s + (0.0330 − 0.504i)9-s + (−3.14 + 2.97i)10-s + (−2.38 − 3.82i)11-s + (3.12 + 0.481i)12-s + (−0.257 − 2.61i)13-s + (0.104 − 3.74i)14-s + (−1.84 + 4.46i)15-s + (0.952 − 3.88i)16-s + (−3.66 − 0.482i)17-s + ⋯ |
| L(s) = 1 | + (−0.326 − 0.945i)2-s + (−0.623 − 0.665i)3-s + (−0.786 + 0.617i)4-s + (−0.564 − 1.24i)5-s + (−0.425 + 0.806i)6-s + (0.935 + 0.352i)7-s + (0.840 + 0.542i)8-s + (0.0110 − 0.168i)9-s + (−0.993 + 0.940i)10-s + (−0.717 − 1.15i)11-s + (0.901 + 0.138i)12-s + (−0.0714 − 0.725i)13-s + (0.0278 − 0.999i)14-s + (−0.477 + 1.15i)15-s + (0.238 − 0.971i)16-s + (−0.888 − 0.116i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.178 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.178 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.360103 + 0.431512i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.360103 + 0.431512i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.461 + 1.33i)T \) |
| 7 | \( 1 + (-2.47 - 0.933i)T \) |
| good | 3 | \( 1 + (1.07 + 1.15i)T + (-0.196 + 2.99i)T^{2} \) |
| 5 | \( 1 + (1.26 + 2.78i)T + (-3.29 + 3.75i)T^{2} \) |
| 11 | \( 1 + (2.38 + 3.82i)T + (-4.86 + 9.86i)T^{2} \) |
| 13 | \( 1 + (0.257 + 2.61i)T + (-12.7 + 2.53i)T^{2} \) |
| 17 | \( 1 + (3.66 + 0.482i)T + (16.4 + 4.39i)T^{2} \) |
| 19 | \( 1 + (-0.530 + 3.21i)T + (-17.9 - 6.10i)T^{2} \) |
| 23 | \( 1 + (-2.82 - 3.22i)T + (-3.00 + 22.8i)T^{2} \) |
| 29 | \( 1 + (2.48 + 4.65i)T + (-16.1 + 24.1i)T^{2} \) |
| 31 | \( 1 + (-1.08 + 0.291i)T + (26.8 - 15.5i)T^{2} \) |
| 37 | \( 1 + (1.54 - 4.11i)T + (-27.8 - 24.3i)T^{2} \) |
| 41 | \( 1 + (-0.787 - 3.96i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (2.10 - 0.639i)T + (35.7 - 23.8i)T^{2} \) |
| 47 | \( 1 + (-8.00 + 10.4i)T + (-12.1 - 45.3i)T^{2} \) |
| 53 | \( 1 + (-6.40 - 10.3i)T + (-23.4 + 47.5i)T^{2} \) |
| 59 | \( 1 + (-1.60 + 2.23i)T + (-18.9 - 55.8i)T^{2} \) |
| 61 | \( 1 + (3.94 - 0.919i)T + (54.7 - 26.9i)T^{2} \) |
| 67 | \( 1 + (-6.36 - 6.79i)T + (-4.38 + 66.8i)T^{2} \) |
| 71 | \( 1 + (3.14 + 4.70i)T + (-27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (4.31 - 2.12i)T + (44.4 - 57.9i)T^{2} \) |
| 79 | \( 1 + (-1.93 - 14.6i)T + (-76.3 + 20.4i)T^{2} \) |
| 83 | \( 1 + (-2.43 + 2.96i)T + (-16.1 - 81.4i)T^{2} \) |
| 89 | \( 1 + (1.79 + 5.28i)T + (-70.6 + 54.1i)T^{2} \) |
| 97 | \( 1 + (7.42 - 7.42i)T - 97iT^{2} \) |
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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
−9.367951114371500232594423274845, −8.617613280903069121219978806670, −8.139401824003417995190308923813, −7.24442107081044115557042415504, −5.64881591687526388833031301835, −5.08116960449590128438780698254, −4.06327926195796097974525808989, −2.69665663749749710719603221760, −1.22475633372889686670103561265, −0.36998306959658737998012392625,
2.08483852641111932408657586539, 3.98769290555603239608807910897, 4.64956555295955348118975489258, 5.43812757682850672202463024865, 6.67687237910539805314534962259, 7.30156326241441902854835060500, 7.900088661231503614688670703316, 9.000719915747448133884242633841, 10.06760523753713705747492428026, 10.75189340900292200505388098899
