L(s) = 1 | − 4.69i·3-s + (23.4 − 50.7i)5-s + 10.2i·7-s + 220.·9-s − 596.·11-s − 420. i·13-s + (−238. − 109. i)15-s − 974. i·17-s − 380.·19-s + 48.1·21-s − 3.54e3i·23-s + (−2.02e3 − 2.37e3i)25-s − 2.17e3i·27-s − 5.44e3·29-s + 3.62e3·31-s + ⋯ |
L(s) = 1 | − 0.301i·3-s + (0.419 − 0.907i)5-s + 0.0791i·7-s + 0.909·9-s − 1.48·11-s − 0.690i·13-s + (−0.273 − 0.126i)15-s − 0.817i·17-s − 0.241·19-s + 0.0238·21-s − 1.39i·23-s + (−0.648 − 0.760i)25-s − 0.574i·27-s − 1.20·29-s + 0.677·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.419 + 0.907i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.419 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.817024 - 1.27687i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.817024 - 1.27687i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-23.4 + 50.7i)T \) |
good | 3 | \( 1 + 4.69iT - 243T^{2} \) |
| 7 | \( 1 - 10.2iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 596.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 420. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 974. iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 380.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.54e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 5.44e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.62e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.75e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 263.T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.44e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 2.34e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 3.34e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 2.90e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.94e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 7.16e3iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 8.13e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.51e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 1.64e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.16e5iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 9.93e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 6.29e4iT - 8.58e9T^{2} \) |
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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
−12.92679416449047208180764235359, −12.40453483702343661375675307057, −10.70700921893929268575360738911, −9.746676775657276424557719534153, −8.422607649500975931381776603331, −7.35262407258407877505401068460, −5.70826850914543106217638717547, −4.57640843472720311246107917596, −2.39190260574833468501553529539, −0.63267455149793261088140427788,
2.02433941547405145491299286035, 3.70248155472038926440876750915, 5.34581042469028936698343582556, 6.78346968240514803201157921918, 7.88534418861101424777025832456, 9.604719973938249434703557091211, 10.37343205532108628452901796643, 11.32093331136766734985575120598, 12.91380530354117657122458628456, 13.65489739520135323114390585567