| L(s) = 1 | + (−1.39 + 0.235i)2-s + (1.39 + 0.977i)3-s + (1.88 − 0.655i)4-s + (−1.14 − 1.91i)5-s + (−2.17 − 1.03i)6-s + (−2.67 + 0.717i)7-s + (−2.48 + 1.35i)8-s + (−0.0322 − 0.0887i)9-s + (2.05 + 2.40i)10-s + (4.23 − 2.44i)11-s + (3.27 + 0.931i)12-s + (4.94 − 3.46i)13-s + (3.56 − 1.62i)14-s + (0.272 − 3.80i)15-s + (3.13 − 2.47i)16-s + (−2.15 + 1.00i)17-s + ⋯ |
| L(s) = 1 | + (−0.986 + 0.166i)2-s + (0.806 + 0.564i)3-s + (0.944 − 0.327i)4-s + (−0.513 − 0.858i)5-s + (−0.888 − 0.422i)6-s + (−1.01 + 0.271i)7-s + (−0.877 + 0.480i)8-s + (−0.0107 − 0.0295i)9-s + (0.649 + 0.760i)10-s + (1.27 − 0.736i)11-s + (0.946 + 0.268i)12-s + (1.37 − 0.960i)13-s + (0.952 − 0.435i)14-s + (0.0703 − 0.981i)15-s + (0.784 − 0.619i)16-s + (−0.522 + 0.243i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.870 + 0.492i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.870 + 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.976350 - 0.256967i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.976350 - 0.256967i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.39 - 0.235i)T \) |
| 5 | \( 1 + (1.14 + 1.91i)T \) |
| 19 | \( 1 + (-4.33 - 0.423i)T \) |
| good | 3 | \( 1 + (-1.39 - 0.977i)T + (1.02 + 2.81i)T^{2} \) |
| 7 | \( 1 + (2.67 - 0.717i)T + (6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (-4.23 + 2.44i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-4.94 + 3.46i)T + (4.44 - 12.2i)T^{2} \) |
| 17 | \( 1 + (2.15 - 1.00i)T + (10.9 - 13.0i)T^{2} \) |
| 23 | \( 1 + (-1.25 + 0.109i)T + (22.6 - 3.99i)T^{2} \) |
| 29 | \( 1 + (3.16 + 8.70i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-4.28 - 2.47i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.47 - 1.47i)T - 37iT^{2} \) |
| 41 | \( 1 + (-0.696 - 3.95i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-0.0647 + 0.740i)T + (-42.3 - 7.46i)T^{2} \) |
| 47 | \( 1 + (0.197 + 0.0919i)T + (30.2 + 36.0i)T^{2} \) |
| 53 | \( 1 + (0.555 + 6.35i)T + (-52.1 + 9.20i)T^{2} \) |
| 59 | \( 1 + (2.36 + 0.860i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.194 - 0.163i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-10.3 - 4.81i)T + (43.0 + 51.3i)T^{2} \) |
| 71 | \( 1 + (6.76 + 8.06i)T + (-12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (8.73 - 12.4i)T + (-24.9 - 68.5i)T^{2} \) |
| 79 | \( 1 + (-1.19 - 6.77i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-3.39 - 12.6i)T + (-71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (2.20 + 0.389i)T + (83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (-13.7 + 6.41i)T + (62.3 - 74.3i)T^{2} \) |
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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
−11.23639582015225638850674916854, −9.907443612847897993655871618694, −9.311089166551207824889044172790, −8.606499693528137048234448212237, −8.087667471984975914184020543631, −6.55384310393232507307346799522, −5.77324281627058468577741638980, −3.85558576128593331774968133772, −3.11103868102342013882772879069, −0.939213555351193609726265957842,
1.61053880204858936869550515175, 3.01857179670208447732467464597, 3.84027360222577720010210565771, 6.39965347337872144214295001902, 6.93231690966528928442941908325, 7.61258427721903940382486344279, 8.894716315365554254533169778844, 9.284242206465200380435053190717, 10.47928965006189028224763887590, 11.32723701949484587551476582218
