| L(s) = 1 | + (−0.0159 + 0.00280i)2-s + (−1.87 + 0.683i)4-s + (−3.75 + 1.36i)5-s + (0.157 + 2.64i)7-s + (0.0559 − 0.0323i)8-s + (0.0558 − 0.0322i)10-s + (1.76 − 4.84i)11-s + (−0.220 − 0.606i)13-s + (−0.00992 − 0.0415i)14-s + (3.06 − 2.57i)16-s + (−1.69 − 2.93i)17-s + (0.328 + 0.189i)19-s + (6.11 − 5.13i)20-s + (−0.0144 + 0.0821i)22-s + (0.993 + 0.175i)23-s + ⋯ |
| L(s) = 1 | + (−0.0112 + 0.00198i)2-s + (−0.939 + 0.341i)4-s + (−1.67 + 0.610i)5-s + (0.0596 + 0.998i)7-s + (0.0197 − 0.0114i)8-s + (0.0176 − 0.0102i)10-s + (0.531 − 1.46i)11-s + (−0.0612 − 0.168i)13-s + (−0.00265 − 0.0111i)14-s + (0.765 − 0.642i)16-s + (−0.410 − 0.711i)17-s + (0.0753 + 0.0435i)19-s + (1.36 − 1.14i)20-s + (−0.00308 + 0.0175i)22-s + (0.207 + 0.0365i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.124 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.124 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.308749 - 0.272520i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.308749 - 0.272520i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + (-0.157 - 2.64i)T \) |
| good | 2 | \( 1 + (0.0159 - 0.00280i)T + (1.87 - 0.684i)T^{2} \) |
| 5 | \( 1 + (3.75 - 1.36i)T + (3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 + (-1.76 + 4.84i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (0.220 + 0.606i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (1.69 + 2.93i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.328 - 0.189i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.993 - 0.175i)T + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (1.97 - 5.41i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (2.20 + 6.06i)T + (-23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 - 6.16T + 37T^{2} \) |
| 41 | \( 1 + (0.266 - 0.0969i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (1.45 + 8.22i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (6.91 + 2.51i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (1.71 + 0.992i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.62 - 2.20i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (2.75 - 7.57i)T + (-46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-1.63 + 9.29i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (8.57 + 4.94i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 7.08iT - 73T^{2} \) |
| 79 | \( 1 + (0.222 + 1.26i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-8.07 - 2.93i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (-2.67 + 4.63i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (7.39 - 1.30i)T + (91.1 - 33.1i)T^{2} \) |
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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
−10.81218007805149971823156342491, −9.367632910040113472726058074148, −8.671706774435406961688449372085, −8.056844362449001389329873842598, −7.15327282695289654887942053018, −5.89249215152616189372380041549, −4.75363412296977759182215716017, −3.67033327147807687821298126078, −3.01749186929560685456130278169, −0.28381553461168915277830841111,
1.24533841363929513263138102867, 3.72064346217548242500242779483, 4.34622685042873174766889268968, 4.87952067636290174410080438314, 6.60984731068203096588346016740, 7.59197277409546058218261299056, 8.199063145948390731233806588239, 9.196563084726515392399487913758, 9.971188450058325043572722894059, 10.99125054794890564793475897965
