| L(s) = 1 | + 7.34i·2-s − 38·4-s + 14.6i·5-s + 17·7-s − 161. i·8-s − 108·10-s + 161. i·11-s + 95·13-s + 124. i·14-s + 580.·16-s + 308. i·17-s + 209·19-s − 558. i·20-s − 1.18e3·22-s − 867. i·23-s + ⋯ |
| L(s) = 1 | + 1.83i·2-s − 2.37·4-s + 0.587i·5-s + 0.346·7-s − 2.52i·8-s − 1.08·10-s + 1.33i·11-s + 0.562·13-s + 0.637i·14-s + 2.26·16-s + 1.06i·17-s + 0.578·19-s − 1.39i·20-s − 2.45·22-s − 1.63i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.14781i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.14781i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 7.34iT - 16T^{2} \) |
| 5 | \( 1 - 14.6iT - 625T^{2} \) |
| 7 | \( 1 - 17T + 2.40e3T^{2} \) |
| 11 | \( 1 - 161. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 95T + 2.85e4T^{2} \) |
| 17 | \( 1 - 308. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 209T + 1.30e5T^{2} \) |
| 23 | \( 1 + 867. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 323. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 950T + 9.23e5T^{2} \) |
| 37 | \( 1 + 1.17e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 2.14e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 1.43e3T + 3.41e6T^{2} \) |
| 47 | \( 1 - 1.57e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 2.90e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 2.13e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 1.44e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 3.49e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 1.93e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 9.02e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 5.27e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 6.14e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 1.11e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 2.80e3T + 8.85e7T^{2} \) |
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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
−17.12432981919685863795086223002, −15.78597545736317246807215294290, −14.90261983208475283173188195780, −14.08087565435155013744956907380, −12.60414387217061506931505873307, −10.33359541041103543273828866575, −8.708844452389173878328741915741, −7.40831841059815501859738818531, −6.24959575414367357224578006561, −4.52684156103309417973165359126,
1.05118465924749545793646867202, 3.30149216670597581976194195172, 5.12930890024279187535300812097, 8.429041909226720574549807435556, 9.551644204994504609191490054834, 11.07566588439315144152296955885, 11.81598166515821482872491228168, 13.24534142734715698563260044378, 13.99026056634987009935016140466, 16.14233121629841574068007163944
