| L(s) = 1 | + (8.40 − 4.04i)2-s + (−6.82 − 8.55i)3-s + (34.3 − 43.0i)4-s + (14.9 − 7.18i)5-s + (−92.0 − 44.3i)6-s + (−4.02 − 5.04i)7-s + (47.8 − 209. i)8-s + (27.4 − 120. i)9-s + (96.3 − 120. i)10-s + (77.6 + 340. i)11-s − 602.·12-s + (156. + 684. i)13-s + (−54.2 − 26.1i)14-s + (−163. − 78.6i)15-s + (−54.5 − 239. i)16-s + 833.·17-s + ⋯ |
| L(s) = 1 | + (1.48 − 0.715i)2-s + (−0.437 − 0.549i)3-s + (1.07 − 1.34i)4-s + (0.266 − 0.128i)5-s + (−1.04 − 0.502i)6-s + (−0.0310 − 0.0389i)7-s + (0.264 − 1.15i)8-s + (0.112 − 0.494i)9-s + (0.304 − 0.382i)10-s + (0.193 + 0.847i)11-s − 1.20·12-s + (0.256 + 1.12i)13-s + (−0.0739 − 0.0356i)14-s + (−0.187 − 0.0902i)15-s + (−0.0532 − 0.233i)16-s + 0.699·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0570 + 0.998i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.0570 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.10303 - 1.98625i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.10303 - 1.98625i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + (2.56e3 + 3.73e3i)T \) |
| good | 2 | \( 1 + (-8.40 + 4.04i)T + (19.9 - 25.0i)T^{2} \) |
| 3 | \( 1 + (6.82 + 8.55i)T + (-54.0 + 236. i)T^{2} \) |
| 5 | \( 1 + (-14.9 + 7.18i)T + (1.94e3 - 2.44e3i)T^{2} \) |
| 7 | \( 1 + (4.02 + 5.04i)T + (-3.73e3 + 1.63e4i)T^{2} \) |
| 11 | \( 1 + (-77.6 - 340. i)T + (-1.45e5 + 6.98e4i)T^{2} \) |
| 13 | \( 1 + (-156. - 684. i)T + (-3.34e5 + 1.61e5i)T^{2} \) |
| 17 | \( 1 - 833.T + 1.41e6T^{2} \) |
| 19 | \( 1 + (-449. + 563. i)T + (-5.50e5 - 2.41e6i)T^{2} \) |
| 23 | \( 1 + (290. + 139. i)T + (4.01e6 + 5.03e6i)T^{2} \) |
| 31 | \( 1 + (3.10e3 - 1.49e3i)T + (1.78e7 - 2.23e7i)T^{2} \) |
| 37 | \( 1 + (-2.51e3 + 1.10e4i)T + (-6.24e7 - 3.00e7i)T^{2} \) |
| 41 | \( 1 + 704.T + 1.15e8T^{2} \) |
| 43 | \( 1 + (2.00e4 + 9.66e3i)T + (9.16e7 + 1.14e8i)T^{2} \) |
| 47 | \( 1 + (-4.11e3 - 1.80e4i)T + (-2.06e8 + 9.95e7i)T^{2} \) |
| 53 | \( 1 + (2.71e3 - 1.30e3i)T + (2.60e8 - 3.26e8i)T^{2} \) |
| 59 | \( 1 - 291.T + 7.14e8T^{2} \) |
| 61 | \( 1 + (-7.30e3 - 9.16e3i)T + (-1.87e8 + 8.23e8i)T^{2} \) |
| 67 | \( 1 + (4.56e3 - 1.99e4i)T + (-1.21e9 - 5.85e8i)T^{2} \) |
| 71 | \( 1 + (9.88e3 + 4.33e4i)T + (-1.62e9 + 7.82e8i)T^{2} \) |
| 73 | \( 1 + (-7.31e4 - 3.52e4i)T + (1.29e9 + 1.62e9i)T^{2} \) |
| 79 | \( 1 + (1.79e4 - 7.87e4i)T + (-2.77e9 - 1.33e9i)T^{2} \) |
| 83 | \( 1 + (3.06e4 - 3.84e4i)T + (-8.76e8 - 3.84e9i)T^{2} \) |
| 89 | \( 1 + (7.22e4 - 3.47e4i)T + (3.48e9 - 4.36e9i)T^{2} \) |
| 97 | \( 1 + (-9.86e4 + 1.23e5i)T + (-1.91e9 - 8.37e9i)T^{2} \) |
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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
−15.31512249481081924216548122495, −14.20773131155467255142164201911, −13.10991444237411047855446715660, −12.17162638294385645617961850641, −11.33998313036204947735357119468, −9.590139821719991899301703151301, −6.93674785325384775876633505899, −5.58614767155833683746313245344, −3.93571988612543596873386712245, −1.74988654196608591380697653207,
3.44055413548265904956861703746, 5.13753254554390458370772179847, 6.07977120413346859691655206606, 7.88134840231612164946204001244, 10.19446969216532044625364258261, 11.59974843601870333107715651453, 13.00971063068229068692557721283, 13.95751784072149085064818649785, 15.11523972956412812482063517488, 16.15003606323599626134997626902
