| L(s) = 1 | + (5.89 − 3.40i)2-s + (14.3 − 6.06i)3-s + (7.14 − 12.3i)4-s + (7.45 + 12.9i)5-s + (63.9 − 84.5i)6-s + (−125. − 34.0i)7-s + 120. i·8-s + (169. − 174. i)9-s + (87.8 + 50.7i)10-s + (−83.2 − 48.0i)11-s + (27.5 − 221. i)12-s + 416. i·13-s + (−852. + 225. i)14-s + (185. + 140. i)15-s + (638. + 1.10e3i)16-s + (104. − 180. i)17-s + ⋯ |
| L(s) = 1 | + (1.04 − 0.601i)2-s + (0.921 − 0.389i)3-s + (0.223 − 0.386i)4-s + (0.133 + 0.231i)5-s + (0.725 − 0.959i)6-s + (−0.964 − 0.262i)7-s + 0.665i·8-s + (0.697 − 0.716i)9-s + (0.277 + 0.160i)10-s + (−0.207 − 0.119i)11-s + (0.0552 − 0.443i)12-s + 0.684i·13-s + (−1.16 + 0.306i)14-s + (0.212 + 0.160i)15-s + (0.623 + 1.08i)16-s + (0.0874 − 0.151i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.706 + 0.707i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.706 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.46083 - 1.02085i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.46083 - 1.02085i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-14.3 + 6.06i)T \) |
| 7 | \( 1 + (125. + 34.0i)T \) |
| good | 2 | \( 1 + (-5.89 + 3.40i)T + (16 - 27.7i)T^{2} \) |
| 5 | \( 1 + (-7.45 - 12.9i)T + (-1.56e3 + 2.70e3i)T^{2} \) |
| 11 | \( 1 + (83.2 + 48.0i)T + (8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 - 416. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + (-104. + 180. i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 19 | \( 1 + (2.29e3 - 1.32e3i)T + (1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-2.26e3 + 1.30e3i)T + (3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + 7.22e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + (-3.72e3 - 2.14e3i)T + (1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + (1.83e3 + 3.18e3i)T + (-3.46e7 + 6.00e7i)T^{2} \) |
| 41 | \( 1 + 1.54e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 6.01e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (1.95e3 + 3.38e3i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 + (-3.14e4 - 1.81e4i)T + (2.09e8 + 3.62e8i)T^{2} \) |
| 59 | \( 1 + (3.91e3 - 6.77e3i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-1.74e4 + 1.00e4i)T + (4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (2.19e4 - 3.80e4i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 + 2.60e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (-4.40e4 - 2.54e4i)T + (1.03e9 + 1.79e9i)T^{2} \) |
| 79 | \( 1 + (-2.50e4 - 4.34e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + 1.07e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + (-6.83e4 - 1.18e5i)T + (-2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 + 2.51e4iT - 8.58e9T^{2} \) |
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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
−16.87955957243452117608183015166, −15.13896914428904253368424063397, −14.00622408845308145494608755606, −13.14986532650657516139115890474, −12.14762421499288834396004595011, −10.26838963955757501301722487094, −8.517213971636027573970391696748, −6.54800315462442440732123804346, −4.03342770599586686091509841879, −2.53994537047490986127260456043,
3.28192191034614647507587498039, 5.05292790196677733653101016362, 6.88002193761342838335401757764, 8.871932356020472894309403042600, 10.24726721429747692123873636546, 12.84606632161757755058425871894, 13.35025066226343898933859653926, 14.91711932044620333762534490864, 15.46932272816175551412319179502, 16.72710633732773704114867917139
