| L(s) = 1 | + 29.2i·3-s − 44.5i·7-s − 611.·9-s + 349.·11-s − 255. i·13-s + 1.50e3i·17-s + 2.43e3·19-s + 1.30e3·21-s + 1.43e3i·23-s − 1.07e4i·27-s − 2.87e3·29-s + 8.94e3·31-s + 1.02e4i·33-s − 1.45e4i·37-s + 7.45e3·39-s + ⋯ |
| L(s) = 1 | + 1.87i·3-s − 0.343i·7-s − 2.51·9-s + 0.870·11-s − 0.418i·13-s + 1.26i·17-s + 1.54·19-s + 0.643·21-s + 0.565i·23-s − 2.84i·27-s − 0.634·29-s + 1.67·31-s + 1.63i·33-s − 1.74i·37-s + 0.784·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.184068783\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.184068783\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 29.2iT - 243T^{2} \) |
| 7 | \( 1 + 44.5iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 349.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 255. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 1.50e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 2.43e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.43e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 2.87e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 8.94e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.45e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 7.50e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.34e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 8.44e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 2.83e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 3.53e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.56e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.98e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 6.00e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.92e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 981.T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.83e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 3.94e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 5.13e4iT - 8.58e9T^{2} \) |
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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
−9.900745758678500855330814889987, −9.240931867692437551165514825866, −8.475561946608844354096827158576, −7.42931575780905507382610959423, −6.00656867692833915446715139193, −5.41014122016255727109052770792, −4.23757069724866137810466763143, −3.79310947180989988430143236304, −2.78907783381653592892885495364, −1.01107683512224203869520054938,
0.54196711169644864158081232942, 1.29046248604271250787119040660, 2.35720113877652023978970614333, 3.22819118210796079009359380733, 4.89240732239863293208202858172, 5.92019314678870811777262548437, 6.71902971809780413177019333188, 7.27766551098748351586247086024, 8.150921933402585613712468702966, 8.955198671372483662720698853887
