| L(s) = 1 | + (−1.33 + 0.467i)2-s + 3.43i·3-s + (1.56 − 1.24i)4-s + (−1.60 − 4.58i)6-s + (−1.50 + 2.39i)8-s − 8.79·9-s + (4.28 + 5.36i)12-s + 4.88·13-s + (0.879 − 3.90i)16-s + (11.7 − 4.11i)18-s + 4.79i·23-s + (−8.23 − 5.15i)24-s + 5·25-s + (−6.51 + 2.28i)26-s − 19.8i·27-s + ⋯ |
| L(s) = 1 | + (−0.943 + 0.330i)2-s + 1.98i·3-s + (0.780 − 0.624i)4-s + (−0.656 − 1.87i)6-s + (−0.530 + 0.847i)8-s − 2.93·9-s + (1.23 + 1.54i)12-s + 1.35·13-s + (0.219 − 0.975i)16-s + (2.76 − 0.969i)18-s + 0.999i·23-s + (−1.68 − 1.05i)24-s + 25-s + (−1.27 + 0.448i)26-s − 3.82i·27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.624 - 0.780i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.624 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.281305 + 0.585138i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.281305 + 0.585138i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.33 - 0.467i)T \) |
| 23 | \( 1 - 4.79iT \) |
| good | 3 | \( 1 - 3.43iT - 3T^{2} \) |
| 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 4.88T + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 29 | \( 1 - 6.70T + 29T^{2} \) |
| 31 | \( 1 + 0.309iT - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 3.97T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 6.55iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 9.59iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 14.0iT - 71T^{2} \) |
| 73 | \( 1 + 7.61T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−14.91638966939599024660971052953, −13.89847599594457836917453140912, −11.63646185338952648146025425251, −10.83959491333643579489377414774, −10.05116791121631881129843306158, −9.050555347175065051278580895663, −8.297796315210644879231156661147, −6.24743141901396155617083248304, −5.01967028742978629934617855274, −3.33978666997632839420065090311,
1.21001969050650181210683268297, 2.83912621843438152378424633431, 6.17412101188377704213868847961, 6.95118490259484901301505413767, 8.189106001821206401409176061943, 8.777575588714648881410430401214, 10.69689089889013528641145737225, 11.64277564489542386763297187585, 12.55227948789289119506370890558, 13.32103908567233076052346713733
