| L(s) = 1 | + (−0.951 − 0.309i)2-s + (0.303 − 2.98i)3-s + (−2.42 − 1.76i)4-s + (4.16 − 1.35i)5-s + (−1.21 + 2.74i)6-s + (1.73 + 1.26i)7-s + (4.11 + 5.66i)8-s + (−8.81 − 1.81i)9-s − 4.38·10-s + (9.06 + 6.23i)11-s + (−6 + 6.70i)12-s + (−2.70 + 8.33i)13-s + (−1.26 − 1.73i)14-s + (−2.77 − 12.8i)15-s + (1.54 + 4.75i)16-s + (−5.70 + 1.85i)17-s + ⋯ |
| L(s) = 1 | + (−0.475 − 0.154i)2-s + (0.101 − 0.994i)3-s + (−0.606 − 0.440i)4-s + (0.833 − 0.270i)5-s + (−0.201 + 0.457i)6-s + (0.248 + 0.180i)7-s + (0.514 + 0.707i)8-s + (−0.979 − 0.201i)9-s − 0.438·10-s + (0.823 + 0.566i)11-s + (−0.5 + 0.559i)12-s + (−0.208 + 0.641i)13-s + (−0.0900 − 0.124i)14-s + (−0.185 − 0.856i)15-s + (0.0965 + 0.297i)16-s + (−0.335 + 0.109i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.280 + 0.959i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.280 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.666081 - 0.499416i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.666081 - 0.499416i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.303 + 2.98i)T \) |
| 11 | \( 1 + (-9.06 - 6.23i)T \) |
| good | 2 | \( 1 + (0.951 + 0.309i)T + (3.23 + 2.35i)T^{2} \) |
| 5 | \( 1 + (-4.16 + 1.35i)T + (20.2 - 14.6i)T^{2} \) |
| 7 | \( 1 + (-1.73 - 1.26i)T + (15.1 + 46.6i)T^{2} \) |
| 13 | \( 1 + (2.70 - 8.33i)T + (-136. - 99.3i)T^{2} \) |
| 17 | \( 1 + (5.70 - 1.85i)T + (233. - 169. i)T^{2} \) |
| 19 | \( 1 + (-25.1 + 18.2i)T + (111. - 343. i)T^{2} \) |
| 23 | \( 1 + 26.4iT - 529T^{2} \) |
| 29 | \( 1 + (27.8 - 38.3i)T + (-259. - 799. i)T^{2} \) |
| 31 | \( 1 + (9.51 - 29.2i)T + (-777. - 564. i)T^{2} \) |
| 37 | \( 1 + (23.9 + 17.3i)T + (423. + 1.30e3i)T^{2} \) |
| 41 | \( 1 + (-25.6 - 35.2i)T + (-519. + 1.59e3i)T^{2} \) |
| 43 | \( 1 + 39.7T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-2.52 - 3.47i)T + (-682. + 2.10e3i)T^{2} \) |
| 53 | \( 1 + (10.7 + 3.48i)T + (2.27e3 + 1.65e3i)T^{2} \) |
| 59 | \( 1 + (-19.4 + 26.7i)T + (-1.07e3 - 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-8.45 - 26.0i)T + (-3.01e3 + 2.18e3i)T^{2} \) |
| 67 | \( 1 - 70.7T + 4.48e3T^{2} \) |
| 71 | \( 1 + (-83.8 + 27.2i)T + (4.07e3 - 2.96e3i)T^{2} \) |
| 73 | \( 1 + (50.1 + 36.4i)T + (1.64e3 + 5.06e3i)T^{2} \) |
| 79 | \( 1 + (19.4 - 59.9i)T + (-5.04e3 - 3.66e3i)T^{2} \) |
| 83 | \( 1 + (104. - 34.0i)T + (5.57e3 - 4.04e3i)T^{2} \) |
| 89 | \( 1 - 9.66iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-30.9 + 95.2i)T + (-7.61e3 - 5.53e3i)T^{2} \) |
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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
−16.79321809177892637229277715979, −14.60791862222416971548134086643, −13.88681752722467037357096727701, −12.74103743944564160687099647582, −11.31020867501045699497355934325, −9.555947831618934172423398259989, −8.731486895326381753370355999024, −6.90974430786289162653595320483, −5.21606029326082297504578490767, −1.66830327165204347502839971669,
3.72572575181494395316665160529, 5.62102800622244079470346588353, 7.87081370407600450170214572415, 9.324948714084666073712835276006, 10.01017120487270444435141713211, 11.57212904269183578038756557193, 13.50224018694749957446608842328, 14.28633758365078312504023074910, 15.76414304656259947234741246510, 17.00833510002996825739918816893
