| L(s) = 1 | + (0.796 + 0.605i)2-s + (−0.468 − 0.883i)3-s + (0.267 + 0.963i)4-s + (−2.86 − 2.71i)5-s + (0.161 − 0.986i)6-s + (−2.64 + 3.11i)7-s + (−0.370 + 0.928i)8-s + (−0.561 + 0.827i)9-s + (−0.638 − 3.89i)10-s + (−3.87 − 2.33i)11-s + (0.725 − 0.687i)12-s + (−1.12 − 1.66i)13-s + (−3.99 + 0.878i)14-s + (−1.05 + 3.80i)15-s + (−0.856 + 0.515i)16-s + (2.21 + 2.60i)17-s + ⋯ |
| L(s) = 1 | + (0.562 + 0.427i)2-s + (−0.270 − 0.510i)3-s + (0.133 + 0.481i)4-s + (−1.28 − 1.21i)5-s + (0.0660 − 0.402i)6-s + (−1.00 + 1.17i)7-s + (−0.130 + 0.328i)8-s + (−0.187 + 0.275i)9-s + (−0.201 − 1.23i)10-s + (−1.16 − 0.703i)11-s + (0.209 − 0.198i)12-s + (−0.313 − 0.461i)13-s + (−1.06 + 0.234i)14-s + (−0.272 + 0.981i)15-s + (−0.214 + 0.128i)16-s + (0.536 + 0.631i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 + 0.368i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.929 + 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0423886 - 0.222015i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0423886 - 0.222015i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.796 - 0.605i)T \) |
| 3 | \( 1 + (0.468 + 0.883i)T \) |
| 59 | \( 1 + (-4.06 - 6.51i)T \) |
| good | 5 | \( 1 + (2.86 + 2.71i)T + (0.270 + 4.99i)T^{2} \) |
| 7 | \( 1 + (2.64 - 3.11i)T + (-1.13 - 6.90i)T^{2} \) |
| 11 | \( 1 + (3.87 + 2.33i)T + (5.15 + 9.71i)T^{2} \) |
| 13 | \( 1 + (1.12 + 1.66i)T + (-4.81 + 12.0i)T^{2} \) |
| 17 | \( 1 + (-2.21 - 2.60i)T + (-2.75 + 16.7i)T^{2} \) |
| 19 | \( 1 + (-2.73 + 1.26i)T + (12.3 - 14.4i)T^{2} \) |
| 23 | \( 1 + (8.61 + 2.90i)T + (18.3 + 13.9i)T^{2} \) |
| 29 | \( 1 + (-1.77 + 1.34i)T + (7.75 - 27.9i)T^{2} \) |
| 31 | \( 1 + (3.05 + 1.41i)T + (20.0 + 23.6i)T^{2} \) |
| 37 | \( 1 + (0.550 + 1.38i)T + (-26.8 + 25.4i)T^{2} \) |
| 41 | \( 1 + (-5.40 + 1.82i)T + (32.6 - 24.8i)T^{2} \) |
| 43 | \( 1 + (0.755 - 0.454i)T + (20.1 - 37.9i)T^{2} \) |
| 47 | \( 1 + (-5.27 + 4.99i)T + (2.54 - 46.9i)T^{2} \) |
| 53 | \( 1 + (-1.76 + 10.7i)T + (-50.2 - 16.9i)T^{2} \) |
| 61 | \( 1 + (0.844 + 0.641i)T + (16.3 + 58.7i)T^{2} \) |
| 67 | \( 1 + (5.30 - 13.3i)T + (-48.6 - 46.0i)T^{2} \) |
| 71 | \( 1 + (11.1 - 10.5i)T + (3.84 - 70.8i)T^{2} \) |
| 73 | \( 1 + (10.2 - 2.24i)T + (66.2 - 30.6i)T^{2} \) |
| 79 | \( 1 + (-3.99 + 7.52i)T + (-44.3 - 65.3i)T^{2} \) |
| 83 | \( 1 + (15.2 - 1.65i)T + (81.0 - 17.8i)T^{2} \) |
| 89 | \( 1 + (-1.60 + 1.22i)T + (23.8 - 85.7i)T^{2} \) |
| 97 | \( 1 + (5.08 + 1.11i)T + (88.0 + 40.7i)T^{2} \) |
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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
−11.64210715874115996860477845050, −10.17117454054008484543314066891, −8.747129370173330028842662408819, −8.189662712198948841786078031821, −7.36703833821802524198739040405, −5.82873798822987585126113644882, −5.43699856247318721469348238488, −4.02404046205513945132255782272, −2.75463196677635321882708133749, −0.12570255004386067109295794024,
2.89631733264104708205722085529, 3.71525700510895035230280176809, 4.56005928675852996462301381051, 6.06058042642476928868376666606, 7.28079948862905109693255274797, 7.61166514915848807009610055805, 9.684247587194024926505058595700, 10.26452472434827998099032210476, 10.87820979303921485854966573609, 11.85351852279125145383173270673
