| L(s) = 1 | + (−1.23 − 3.80i)2-s + (−12.9 + 9.40i)4-s + (74.1 + 24.0i)5-s + (−84.1 − 115. i)7-s + (51.7 + 37.6i)8-s − 311. i·10-s + (−369. + 157. i)11-s + (160. − 52.2i)13-s + (−336. + 463. i)14-s + (79.1 − 243. i)16-s + (77.0 − 237. i)17-s + (−448. + 617. i)19-s + (−1.18e3 + 385. i)20-s + (1.05e3 + 1.21e3i)22-s − 4.44e3i·23-s + ⋯ |
| L(s) = 1 | + (−0.218 − 0.672i)2-s + (−0.404 + 0.293i)4-s + (1.32 + 0.430i)5-s + (−0.649 − 0.893i)7-s + (0.286 + 0.207i)8-s − 0.985i·10-s + (−0.920 + 0.391i)11-s + (0.263 − 0.0857i)13-s + (−0.459 + 0.632i)14-s + (0.0772 − 0.237i)16-s + (0.0646 − 0.199i)17-s + (−0.285 + 0.392i)19-s + (−0.662 + 0.215i)20-s + (0.464 + 0.533i)22-s − 1.75i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.905 + 0.423i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.905 + 0.423i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.128060994\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.128060994\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.23 + 3.80i)T \) |
| 3 | \( 1 \) |
| 11 | \( 1 + (369. - 157. i)T \) |
| good | 5 | \( 1 + (-74.1 - 24.0i)T + (2.52e3 + 1.83e3i)T^{2} \) |
| 7 | \( 1 + (84.1 + 115. i)T + (-5.19e3 + 1.59e4i)T^{2} \) |
| 13 | \( 1 + (-160. + 52.2i)T + (3.00e5 - 2.18e5i)T^{2} \) |
| 17 | \( 1 + (-77.0 + 237. i)T + (-1.14e6 - 8.34e5i)T^{2} \) |
| 19 | \( 1 + (448. - 617. i)T + (-7.65e5 - 2.35e6i)T^{2} \) |
| 23 | \( 1 + 4.44e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + (-3.97e3 + 2.88e3i)T + (6.33e6 - 1.95e7i)T^{2} \) |
| 31 | \( 1 + (1.13e3 + 3.47e3i)T + (-2.31e7 + 1.68e7i)T^{2} \) |
| 37 | \( 1 + (1.27e3 - 923. i)T + (2.14e7 - 6.59e7i)T^{2} \) |
| 41 | \( 1 + (1.15e4 + 8.39e3i)T + (3.58e7 + 1.10e8i)T^{2} \) |
| 43 | \( 1 + 5.11e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (2.01e3 - 2.77e3i)T + (-7.08e7 - 2.18e8i)T^{2} \) |
| 53 | \( 1 + (2.87e4 - 9.33e3i)T + (3.38e8 - 2.45e8i)T^{2} \) |
| 59 | \( 1 + (1.66e4 + 2.29e4i)T + (-2.20e8 + 6.79e8i)T^{2} \) |
| 61 | \( 1 + (-2.37e4 - 7.70e3i)T + (6.83e8 + 4.96e8i)T^{2} \) |
| 67 | \( 1 + 5.32e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (4.35e4 + 1.41e4i)T + (1.45e9 + 1.06e9i)T^{2} \) |
| 73 | \( 1 + (-3.02e4 - 4.16e4i)T + (-6.40e8 + 1.97e9i)T^{2} \) |
| 79 | \( 1 + (-1.54e4 + 5.01e3i)T + (2.48e9 - 1.80e9i)T^{2} \) |
| 83 | \( 1 + (-2.28e4 + 7.02e4i)T + (-3.18e9 - 2.31e9i)T^{2} \) |
| 89 | \( 1 + 1.22e5iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (1.69e4 + 5.21e4i)T + (-6.94e9 + 5.04e9i)T^{2} \) |
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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
−10.70302662784015835973200887959, −10.29962653192534759721033421666, −9.602659674518427838766245500841, −8.303494266578423837218734568944, −6.97812832729479789601272180743, −5.97142102919615523595873294774, −4.53610556672007977344354212043, −3.03359664636652242267465621946, −1.97831113465889721463179193391, −0.35537276957528698849973183807,
1.53406026276508784854762539140, 3.01203100474512610140741960624, 5.09415180508327226253549722096, 5.75474582962036919423993427092, 6.65036166777011288826753499650, 8.139708499783287718305422372955, 9.103766384056089443916612557231, 9.721065183288505055953346392598, 10.76896893090807971640656242954, 12.26041707550772728885575255716
