| L(s) = 1 | − 2.79·7-s − 18.1i·11-s − 23.0·13-s − 5.72i·17-s − 23.1·19-s + 0.271i·23-s − 39.7i·29-s + 47.3·31-s + 34.8·37-s + 13.2i·41-s − 46.7·43-s + 40.9i·47-s − 41.1·49-s + 91.3i·53-s + 78.8i·59-s + ⋯ |
| L(s) = 1 | − 0.399·7-s − 1.64i·11-s − 1.77·13-s − 0.336i·17-s − 1.22·19-s + 0.0118i·23-s − 1.37i·29-s + 1.52·31-s + 0.942·37-s + 0.323i·41-s − 1.08·43-s + 0.870i·47-s − 0.840·49-s + 1.72i·53-s + 1.33i·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5598183185\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5598183185\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 2.79T + 49T^{2} \) |
| 11 | \( 1 + 18.1iT - 121T^{2} \) |
| 13 | \( 1 + 23.0T + 169T^{2} \) |
| 17 | \( 1 + 5.72iT - 289T^{2} \) |
| 19 | \( 1 + 23.1T + 361T^{2} \) |
| 23 | \( 1 - 0.271iT - 529T^{2} \) |
| 29 | \( 1 + 39.7iT - 841T^{2} \) |
| 31 | \( 1 - 47.3T + 961T^{2} \) |
| 37 | \( 1 - 34.8T + 1.36e3T^{2} \) |
| 41 | \( 1 - 13.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 46.7T + 1.84e3T^{2} \) |
| 47 | \( 1 - 40.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 91.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 78.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 31.1T + 3.72e3T^{2} \) |
| 67 | \( 1 - 6.91T + 4.48e3T^{2} \) |
| 71 | \( 1 - 81.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 106.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 63.5T + 6.24e3T^{2} \) |
| 83 | \( 1 + 0.284iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 28.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 92.9T + 9.40e3T^{2} \) |
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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
−8.767855543832673991046626284343, −8.135857412336659941704156581786, −7.40340734545038233741663001138, −6.38730148304007399904764959371, −5.96838039937884268618521179063, −4.87453354233001276464875729511, −4.17246129480883757912666248341, −2.96813890919911416334953831523, −2.44373682512722231671092268045, −0.819698412641613826463355534786,
0.15998196583791466762455385625, 1.85400819303385565261695359561, 2.50455012352962447381820174119, 3.67752841367887793725343550825, 4.78763003695275650486338861557, 4.99480002168681386844835383558, 6.48593933064925652448318799389, 6.83023545267924286114123052777, 7.67608349142465871769413468758, 8.411968670331742854307221101289
