| L(s) = 1 | + (−1.12 + 3.46i)2-s + (−12.9 − 8.74i)3-s + (15.1 + 10.9i)4-s + (54.0 − 17.5i)5-s + (44.8 − 34.9i)6-s + (−123. + 170. i)7-s + (−149. + 108. i)8-s + (90.2 + 225. i)9-s + 207. i·10-s + (−371. + 151. i)11-s + (−99.1 − 274. i)12-s + (787. + 255. i)13-s + (−450. − 620. i)14-s + (−850. − 245. i)15-s + (−23.4 − 72.1i)16-s + (271. + 836. i)17-s + ⋯ |
| L(s) = 1 | + (−0.199 + 0.613i)2-s + (−0.828 − 0.560i)3-s + (0.472 + 0.343i)4-s + (0.966 − 0.313i)5-s + (0.508 − 0.395i)6-s + (−0.952 + 1.31i)7-s + (−0.826 + 0.600i)8-s + (0.371 + 0.928i)9-s + 0.655i·10-s + (−0.925 + 0.378i)11-s + (−0.198 − 0.549i)12-s + (1.29 + 0.419i)13-s + (−0.614 − 0.845i)14-s + (−0.976 − 0.281i)15-s + (−0.0229 − 0.0704i)16-s + (0.228 + 0.701i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.388 - 0.921i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.388 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.606656 + 0.914589i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.606656 + 0.914589i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (12.9 + 8.74i)T \) |
| 11 | \( 1 + (371. - 151. i)T \) |
| good | 2 | \( 1 + (1.12 - 3.46i)T + (-25.8 - 18.8i)T^{2} \) |
| 5 | \( 1 + (-54.0 + 17.5i)T + (2.52e3 - 1.83e3i)T^{2} \) |
| 7 | \( 1 + (123. - 170. i)T + (-5.19e3 - 1.59e4i)T^{2} \) |
| 13 | \( 1 + (-787. - 255. i)T + (3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (-271. - 836. i)T + (-1.14e6 + 8.34e5i)T^{2} \) |
| 19 | \( 1 + (-68.7 - 94.5i)T + (-7.65e5 + 2.35e6i)T^{2} \) |
| 23 | \( 1 + 181. iT - 6.43e6T^{2} \) |
| 29 | \( 1 + (2.73e3 + 1.98e3i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-3.13e3 + 9.63e3i)T + (-2.31e7 - 1.68e7i)T^{2} \) |
| 37 | \( 1 + (-7.06e3 - 5.13e3i)T + (2.14e7 + 6.59e7i)T^{2} \) |
| 41 | \( 1 + (-9.18e3 + 6.67e3i)T + (3.58e7 - 1.10e8i)T^{2} \) |
| 43 | \( 1 - 1.32e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (-5.65e3 - 7.78e3i)T + (-7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (1.81e4 + 5.88e3i)T + (3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (1.18e4 - 1.63e4i)T + (-2.20e8 - 6.79e8i)T^{2} \) |
| 61 | \( 1 + (-1.69e4 + 5.50e3i)T + (6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 - 2.18e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (-4.97e4 + 1.61e4i)T + (1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + (-2.26e3 + 3.12e3i)T + (-6.40e8 - 1.97e9i)T^{2} \) |
| 79 | \( 1 + (-1.72e4 - 5.60e3i)T + (2.48e9 + 1.80e9i)T^{2} \) |
| 83 | \( 1 + (-4.03e3 - 1.24e4i)T + (-3.18e9 + 2.31e9i)T^{2} \) |
| 89 | \( 1 + 2.96e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (1.76e3 - 5.44e3i)T + (-6.94e9 - 5.04e9i)T^{2} \) |
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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
−16.18322753614231715717235701935, −15.35860336815112074019336138722, −13.28128277302392941487529797709, −12.55838863429140567479137669890, −11.26991770501170122217159351805, −9.542379706765272012244374277961, −8.031119023356396480356491976100, −6.21944876654766408761244877649, −5.81484244201246306575781489037, −2.25737566651854250584300344471,
0.77192450923203423613476931211, 3.31545947599791825804287391147, 5.72958414188224542178303609174, 6.79319254730811673898494274969, 9.590540792672982685583906824106, 10.45332831424386886775018662890, 10.99913848678815870401179790427, 12.76663914202113944917529599342, 13.91276246750933176699120410212, 15.74211488250979634757843359689
