L(s) = 1 | + (0.707 + 1.22i)2-s + (−0.999 + 1.73i)4-s + (4.52 + 2.12i)5-s + (−1.70 + 0.982i)7-s − 2.82·8-s + (0.594 + 7.04i)10-s + (−3.30 + 1.90i)11-s + (−16.1 − 9.30i)13-s + (−2.40 − 1.38i)14-s + (−2.00 − 3.46i)16-s − 17.6·17-s − 26.0·19-s + (−8.20 + 5.71i)20-s + (−4.67 − 2.69i)22-s + (−4.82 + 8.34i)23-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.905 + 0.425i)5-s + (−0.243 + 0.140i)7-s − 0.353·8-s + (0.0594 + 0.704i)10-s + (−0.300 + 0.173i)11-s + (−1.24 − 0.716i)13-s + (−0.171 − 0.0992i)14-s + (−0.125 − 0.216i)16-s − 1.03·17-s − 1.37·19-s + (−0.410 + 0.285i)20-s + (−0.212 − 0.122i)22-s + (−0.209 + 0.363i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.867 + 0.497i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.867 + 0.497i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6751575601\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6751575601\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 1.22i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-4.52 - 2.12i)T \) |
good | 7 | \( 1 + (1.70 - 0.982i)T + (24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (3.30 - 1.90i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (16.1 + 9.30i)T + (84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + 17.6T + 289T^{2} \) |
| 19 | \( 1 + 26.0T + 361T^{2} \) |
| 23 | \( 1 + (4.82 - 8.34i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-5.20 + 3.00i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (12.6 - 21.9i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 - 39.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-30.7 - 17.7i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-52.8 + 30.5i)T + (924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (35.4 + 61.3i)T + (-1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + 83.3T + 2.80e3T^{2} \) |
| 59 | \( 1 + (32.3 + 18.6i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-2.37 - 4.11i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-8.03 - 4.63i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 34.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 22.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 + (-49.5 - 85.8i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (18.9 + 32.8i)T + (-3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + 43.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-23.7 + 13.7i)T + (4.70e3 - 8.14e3i)T^{2} \) |
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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
−10.40685810519102477106910900652, −9.692077621698212030287290651117, −8.835732788149830954034298704550, −7.84927670178989630207038895999, −6.89720587758566171507247108141, −6.25690499118150446971480305699, −5.31258918193992509141301745146, −4.48315150863688007954188887895, −3.02604841608844331218166233699, −2.12378967966531583029448863706,
0.17766289490119415010789384679, 1.92380653738793808987162272608, 2.60631597702470933508949152783, 4.22418562716954692822513046242, 4.82960335681066237069486118437, 5.99071628522036312480005175523, 6.67362729523071637720446040222, 7.941723127267424096920423014166, 9.182633599862085174924809092800, 9.429221682476884690316320537908