| L(s) = 1 | − 3.55i·2-s − 8.64·4-s + 9.40i·5-s − 2.64·7-s + 16.5i·8-s + 33.4·10-s + 3.55i·11-s + 11.6·13-s + 9.40i·14-s + 24.1·16-s + 27.1i·17-s + 6.52·19-s − 81.3i·20-s + 12.6·22-s + 1.03i·23-s + ⋯ |
| L(s) = 1 | − 1.77i·2-s − 2.16·4-s + 1.88i·5-s − 0.377·7-s + 2.06i·8-s + 3.34·10-s + 0.323i·11-s + 0.895·13-s + 0.672i·14-s + 1.51·16-s + 1.59i·17-s + 0.343·19-s − 4.06i·20-s + 0.574·22-s + 0.0450i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.03031\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.03031\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + 2.64T \) |
| good | 2 | \( 1 + 3.55iT - 4T^{2} \) |
| 5 | \( 1 - 9.40iT - 25T^{2} \) |
| 11 | \( 1 - 3.55iT - 121T^{2} \) |
| 13 | \( 1 - 11.6T + 169T^{2} \) |
| 17 | \( 1 - 27.1iT - 289T^{2} \) |
| 19 | \( 1 - 6.52T + 361T^{2} \) |
| 23 | \( 1 - 1.03iT - 529T^{2} \) |
| 29 | \( 1 - 29.7iT - 841T^{2} \) |
| 31 | \( 1 + 24.3T + 961T^{2} \) |
| 37 | \( 1 + 45.2T + 1.36e3T^{2} \) |
| 41 | \( 1 + 19.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 74.6T + 1.84e3T^{2} \) |
| 47 | \( 1 - 7.92iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 9.63iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 1.70iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 98.4T + 3.72e3T^{2} \) |
| 67 | \( 1 + 27.8T + 4.48e3T^{2} \) |
| 71 | \( 1 + 63.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 33.7T + 5.32e3T^{2} \) |
| 79 | \( 1 + 58.6T + 6.24e3T^{2} \) |
| 83 | \( 1 + 67.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 25.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 18.2T + 9.40e3T^{2} \) |
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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
−12.15781763552038641818258413167, −11.00039910549897188283289166795, −10.69596835691072811900939199908, −9.901836528619403342481387054949, −8.712922005625678938926573025888, −7.18851807216512888061728722536, −5.92803095851490694075688131557, −3.88309497078710048349157427112, −3.21017752843723406380439389958, −1.90906493121477313643922568113,
0.62882790953537701146558604373, 4.09686092634987802290807770820, 5.17795544687596938370658913589, 5.87100154760007751006762718250, 7.22430120390574426413990746844, 8.267307339652795593814267019001, 8.978670741285576359503607000797, 9.637122761009663942800379056683, 11.63120039818727981361429120856, 12.80728321733400213050738987263
