| L(s) = 1 | + (−0.602 − 1.62i)3-s + (−2.89 − 1.05i)5-s + (−0.0578 + 0.328i)7-s + (−2.27 + 1.95i)9-s + (0.0918 − 0.0334i)11-s + (−0.709 + 0.595i)13-s + (0.0324 + 5.33i)15-s + (−1.04 − 1.81i)17-s + (−0.0352 + 0.0609i)19-s + (0.567 − 0.103i)21-s + (1.09 + 6.20i)23-s + (3.42 + 2.87i)25-s + (4.54 + 2.51i)27-s + (−5.42 − 4.55i)29-s + (1.69 + 9.59i)31-s + ⋯ |
| L(s) = 1 | + (−0.347 − 0.937i)3-s + (−1.29 − 0.470i)5-s + (−0.0218 + 0.123i)7-s + (−0.758 + 0.652i)9-s + (0.0276 − 0.0100i)11-s + (−0.196 + 0.165i)13-s + (0.00837 + 1.37i)15-s + (−0.253 − 0.439i)17-s + (−0.00807 + 0.0139i)19-s + (0.123 − 0.0226i)21-s + (0.228 + 1.29i)23-s + (0.685 + 0.575i)25-s + (0.875 + 0.484i)27-s + (−1.00 − 0.844i)29-s + (0.303 + 1.72i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.379 - 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.379 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.361147 + 0.242280i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.361147 + 0.242280i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.602 + 1.62i)T \) |
| good | 5 | \( 1 + (2.89 + 1.05i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (0.0578 - 0.328i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (-0.0918 + 0.0334i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (0.709 - 0.595i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (1.04 + 1.81i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.0352 - 0.0609i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.09 - 6.20i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (5.42 + 4.55i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-1.69 - 9.59i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-2.24 - 3.89i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.632 + 0.530i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-8.73 + 3.17i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (0.949 - 5.38i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 8.63T + 53T^{2} \) |
| 59 | \( 1 + (-13.0 - 4.73i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (0.712 - 4.03i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (9.50 - 7.97i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (2.92 + 5.05i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.07 - 3.59i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4.81 + 4.03i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (11.4 + 9.58i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-1.08 + 1.87i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (2.94 - 1.07i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.54447975340498528201983462719, −9.237812582492722286593213728660, −8.500272798431715369035870142037, −7.57166252640727380808108002827, −7.21279344951100324800367782929, −6.00749121345244611630073864943, −5.05644575478961810398151836172, −4.04137654482445061058379238393, −2.78165481321689285125726217084, −1.23090468245709611280354320862,
0.24521972415327781996704710122, 2.71384263159248833648453880419, 3.85709210401316606563769896351, 4.33557654939451891219748693770, 5.52316623990260694990019274129, 6.54622259888823995027571430784, 7.50489375913230094928348275033, 8.333912337613261764236984138333, 9.221438426765145681464708845734, 10.14475402210544026021619473312
