| L(s) = 1 | + (−0.777 + 1.18i)2-s + (−1.85 + 2.20i)3-s + (−0.790 − 1.83i)4-s + (−0.543 − 2.16i)5-s + (−1.16 − 3.90i)6-s + (0.0958 + 0.165i)7-s + (2.78 + 0.494i)8-s + (−0.920 − 5.22i)9-s + (2.98 + 1.04i)10-s + (0.219 + 0.126i)11-s + (5.51 + 1.65i)12-s + (1.99 − 1.67i)13-s + (−0.270 − 0.0158i)14-s + (5.79 + 2.81i)15-s + (−2.74 + 2.90i)16-s + (2.07 + 0.365i)17-s + ⋯ |
| L(s) = 1 | + (−0.549 + 0.835i)2-s + (−1.06 + 1.27i)3-s + (−0.395 − 0.918i)4-s + (−0.243 − 0.970i)5-s + (−0.476 − 1.59i)6-s + (0.0362 + 0.0627i)7-s + (0.984 + 0.174i)8-s + (−0.306 − 1.74i)9-s + (0.943 + 0.330i)10-s + (0.0663 + 0.0382i)11-s + (1.59 + 0.478i)12-s + (0.553 − 0.464i)13-s + (−0.0723 − 0.00423i)14-s + (1.49 + 0.727i)15-s + (−0.687 + 0.726i)16-s + (0.503 + 0.0887i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.685 - 0.728i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.685 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.584187 + 0.252450i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.584187 + 0.252450i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.777 - 1.18i)T \) |
| 5 | \( 1 + (0.543 + 2.16i)T \) |
| 19 | \( 1 + (-4.07 + 1.54i)T \) |
| good | 3 | \( 1 + (1.85 - 2.20i)T + (-0.520 - 2.95i)T^{2} \) |
| 7 | \( 1 + (-0.0958 - 0.165i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.219 - 0.126i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.99 + 1.67i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.07 - 0.365i)T + (15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (4.63 + 1.68i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-6.52 + 1.15i)T + (27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-1.47 - 2.55i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 6.57T + 37T^{2} \) |
| 41 | \( 1 + (7.06 - 8.41i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-6.67 + 2.42i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (0.936 + 5.31i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-4.80 - 1.74i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (-2.04 + 11.5i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-11.1 - 4.06i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-2.04 + 0.360i)T + (62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-9.44 + 3.43i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-2.30 + 2.74i)T + (-12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (5.82 + 4.88i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-1.59 - 2.76i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (8.40 + 10.0i)T + (-15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-2.46 + 13.9i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−11.33652846813826276049433332208, −10.21398198597158236232281255617, −9.785863887942189227803226704873, −8.750669399230912524561725287606, −7.954334564653848046374285834892, −6.47033123687569479913904026604, −5.53010896574646217819015676341, −4.91692198588860767462155999215, −3.92337822805430839137374170161, −0.77332672403431180537651591182,
1.09771276012533583771834262996, 2.50059938103932678324844950542, 3.94958441900834832002243768234, 5.65072376473483994405277783863, 6.69058536459102008567623113663, 7.48366309576185210570855914282, 8.215520128824704759604511526292, 9.721779220374040861291850731480, 10.60283924690835507320335028771, 11.42541788545950346246600749163
