| L(s) = 1 | + (0.984 − 0.173i)2-s + (−1.64 − 0.540i)3-s + (0.939 − 0.342i)4-s + (0.881 + 0.739i)5-s + (−1.71 − 0.246i)6-s + (−0.336 + 2.62i)7-s + (0.866 − 0.5i)8-s + (2.41 + 1.77i)9-s + (0.996 + 0.575i)10-s + (0.595 + 0.709i)11-s + (−1.73 + 0.0550i)12-s + (4.25 + 0.749i)13-s + (0.124 + 2.64i)14-s + (−1.05 − 1.69i)15-s + (0.766 − 0.642i)16-s + (2.28 − 3.95i)17-s + ⋯ |
| L(s) = 1 | + (0.696 − 0.122i)2-s + (−0.950 − 0.311i)3-s + (0.469 − 0.171i)4-s + (0.394 + 0.330i)5-s + (−0.699 − 0.100i)6-s + (−0.127 + 0.991i)7-s + (0.306 − 0.176i)8-s + (0.805 + 0.592i)9-s + (0.315 + 0.181i)10-s + (0.179 + 0.214i)11-s + (−0.499 + 0.0158i)12-s + (1.17 + 0.207i)13-s + (0.0332 + 0.706i)14-s + (−0.271 − 0.437i)15-s + (0.191 − 0.160i)16-s + (0.553 − 0.958i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.199i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 - 0.199i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.68693 + 0.169716i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.68693 + 0.169716i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.984 + 0.173i)T \) |
| 3 | \( 1 + (1.64 + 0.540i)T \) |
| 7 | \( 1 + (0.336 - 2.62i)T \) |
| good | 5 | \( 1 + (-0.881 - 0.739i)T + (0.868 + 4.92i)T^{2} \) |
| 11 | \( 1 + (-0.595 - 0.709i)T + (-1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-4.25 - 0.749i)T + (12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-2.28 + 3.95i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.687 - 0.397i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.46 - 4.01i)T + (-17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-1.10 + 0.195i)T + (27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-1.11 - 3.07i)T + (-23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (5.27 - 9.13i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.15 + 6.56i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (2.26 - 1.89i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-1.43 - 0.522i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + 6.12iT - 53T^{2} \) |
| 59 | \( 1 + (-0.539 - 0.453i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (2.12 - 5.83i)T + (-46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.0261 + 0.148i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (13.9 + 8.05i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (5.65 - 3.26i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.17 + 12.3i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (2.26 + 12.8i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (6.04 + 10.4i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.91 - 4.66i)T + (-16.8 + 95.5i)T^{2} \) |
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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
−11.81309550467848404638861926350, −10.67822008948551551598700311966, −9.856459179671618537910574877442, −8.635374654040669663412596264744, −7.24157433510216101067823117377, −6.31948968094841739169646279727, −5.67978328996811767079859564251, −4.69981204465554005282071420989, −3.14476002306736864745971061018, −1.66620009950497562630093771019,
1.25300665112811944624295830200, 3.57177506903511634709695011677, 4.37660203840667252278236896740, 5.59018024630883108434756811738, 6.26189627198707309610604929036, 7.23980202863669173929254372575, 8.523277276878144433555056294866, 9.799034357817417917178343508342, 10.72050723090199993885663500553, 11.16743936406412043718252732069
