| L(s) = 1 | + (2.10 + 1.76i)2-s + (0.673 − 1.59i)3-s + (0.963 + 5.46i)4-s + (−0.0619 − 0.351i)5-s + (4.23 − 2.17i)6-s + (−2.30 − 1.30i)7-s + (−4.87 + 8.44i)8-s + (−2.09 − 2.14i)9-s + (0.490 − 0.849i)10-s + (0.309 − 1.75i)11-s + (9.36 + 2.13i)12-s + (−0.143 − 0.811i)13-s + (−2.53 − 6.81i)14-s + (−0.602 − 0.137i)15-s + (−14.7 + 5.36i)16-s + (0.925 − 1.60i)17-s + ⋯ |
| L(s) = 1 | + (1.48 + 1.24i)2-s + (0.388 − 0.921i)3-s + (0.481 + 2.73i)4-s + (−0.0277 − 0.157i)5-s + (1.72 − 0.886i)6-s + (−0.869 − 0.494i)7-s + (−1.72 + 2.98i)8-s + (−0.698 − 0.716i)9-s + (0.155 − 0.268i)10-s + (0.0932 − 0.528i)11-s + (2.70 + 0.617i)12-s + (−0.0397 − 0.225i)13-s + (−0.676 − 1.82i)14-s + (−0.155 − 0.0355i)15-s + (−3.68 + 1.34i)16-s + (0.224 − 0.388i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.468 - 0.883i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.468 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.08289 + 1.25371i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.08289 + 1.25371i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.673 + 1.59i)T \) |
| 7 | \( 1 + (2.30 + 1.30i)T \) |
| good | 2 | \( 1 + (-2.10 - 1.76i)T + (0.347 + 1.96i)T^{2} \) |
| 5 | \( 1 + (0.0619 + 0.351i)T + (-4.69 + 1.71i)T^{2} \) |
| 11 | \( 1 + (-0.309 + 1.75i)T + (-10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (0.143 + 0.811i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-0.925 + 1.60i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.45 - 4.24i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (5.00 - 4.20i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (0.600 - 3.40i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (1.60 + 9.08i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 - 4.98T + 37T^{2} \) |
| 41 | \( 1 + (0.733 + 4.16i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-3.83 - 3.21i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (1.59 - 9.03i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (-2.16 - 3.74i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (6.88 + 2.50i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.822 + 4.66i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-3.34 + 2.80i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-2.19 - 3.80i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 4.69T + 73T^{2} \) |
| 79 | \( 1 + (8.77 + 7.36i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (1.28 - 7.28i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (-2.68 - 4.64i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.12 - 0.939i)T + (16.8 + 95.5i)T^{2} \) |
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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
−12.90908745579598473465420419581, −12.42457271499907898320109273096, −11.39133057497424356760628219441, −9.355602573558757794421326947176, −8.010075580544762602093781176379, −7.44295307999855770078043631886, −6.33543757820774209126828274537, −5.66055716937205845714765826747, −3.94342216538345549014708882556, −2.97055114538836900563140286023,
2.42337986035498680238433412798, 3.39366175550563095293398820933, 4.48383912345489242281893515971, 5.50176431486489671822373921348, 6.68953117440731492415026812705, 8.964095269620660927245586747364, 9.871369170477278136034458823138, 10.52360406602138786684892385784, 11.56338298438764033648496955063, 12.41198944068005122273947934276
