| L(s) = 1 | + 16.9i·3-s − 26.9·5-s + 194.·7-s − 43.2·9-s + 117. i·11-s + 733.·13-s − 456. i·15-s + 1.49e3i·17-s − 67.5i·19-s + 3.29e3i·21-s − 2.88e3·23-s − 2.39e3·25-s + 3.37e3i·27-s + (988. + 4.41e3i)29-s − 7.26e3i·31-s + ⋯ |
| L(s) = 1 | + 1.08i·3-s − 0.482·5-s + 1.50·7-s − 0.178·9-s + 0.292i·11-s + 1.20·13-s − 0.523i·15-s + 1.25i·17-s − 0.0429i·19-s + 1.63i·21-s − 1.13·23-s − 0.767·25-s + 0.892i·27-s + (0.218 + 0.975i)29-s − 1.35i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.218 - 0.975i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.218 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.998876147\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.998876147\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 + (-988. - 4.41e3i)T \) |
| good | 3 | \( 1 - 16.9iT - 243T^{2} \) |
| 5 | \( 1 + 26.9T + 3.12e3T^{2} \) |
| 7 | \( 1 - 194.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 117. iT - 1.61e5T^{2} \) |
| 13 | \( 1 - 733.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.49e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 67.5iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 2.88e3T + 6.43e6T^{2} \) |
| 31 | \( 1 + 7.26e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + 4.91e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.29e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 1.73e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 1.40e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 3.02e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 6.49e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.31e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 8.36e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.32e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.38e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 3.11e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + 4.18e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.10e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + 2.67e4iT - 8.58e9T^{2} \) |
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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
−12.86776541773973074775743943552, −11.48562699968657302212081578547, −10.91953499383843624264084247256, −9.866894298540995900592988983548, −8.534716330574212198016504764231, −7.77841213432731987232178521076, −5.92457492804346017452434041260, −4.53907884201622523487174136821, −3.83219736502222283106690013284, −1.59882095718818615493413618076,
0.818421583261926834397623897669, 2.03727875029393366207293544419, 4.04298991565544232197826511405, 5.53919464522940076516865603571, 6.96944876192569336280703497577, 7.920653246834997238706995536955, 8.637260115504506885788370911111, 10.44223715326502471889475119476, 11.68395809648776123868716082515, 11.99667531334893106988147407486
