| L(s) = 1 | − 7.31i·5-s − 9.37·7-s + 11.7i·11-s + 22.7·13-s − 15.6i·17-s − 18.1·19-s − 23.4i·23-s − 28.4·25-s + 38.7i·29-s − 25.7·31-s + 68.5i·35-s − 2.91·37-s − 40.6i·41-s − 37.9·43-s − 18.8i·47-s + ⋯ |
| L(s) = 1 | − 1.46i·5-s − 1.33·7-s + 1.07i·11-s + 1.75·13-s − 0.918i·17-s − 0.956·19-s − 1.01i·23-s − 1.13·25-s + 1.33i·29-s − 0.831·31-s + 1.95i·35-s − 0.0788·37-s − 0.992i·41-s − 0.883·43-s − 0.400i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4719121928\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4719121928\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 7.31iT - 25T^{2} \) |
| 7 | \( 1 + 9.37T + 49T^{2} \) |
| 11 | \( 1 - 11.7iT - 121T^{2} \) |
| 13 | \( 1 - 22.7T + 169T^{2} \) |
| 17 | \( 1 + 15.6iT - 289T^{2} \) |
| 19 | \( 1 + 18.1T + 361T^{2} \) |
| 23 | \( 1 + 23.4iT - 529T^{2} \) |
| 29 | \( 1 - 38.7iT - 841T^{2} \) |
| 31 | \( 1 + 25.7T + 961T^{2} \) |
| 37 | \( 1 + 2.91T + 1.36e3T^{2} \) |
| 41 | \( 1 + 40.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 37.9T + 1.84e3T^{2} \) |
| 47 | \( 1 + 18.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 12.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 12.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 90.2T + 3.72e3T^{2} \) |
| 67 | \( 1 + 8.73T + 4.48e3T^{2} \) |
| 71 | \( 1 - 94.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 50.6T + 5.32e3T^{2} \) |
| 79 | \( 1 + 102.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 18.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 61.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 148.T + 9.40e3T^{2} \) |
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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
−8.920620996077229594131577363625, −8.279895230000288924904917217412, −7.11872533952388462135417495890, −6.53540814037497817148322337381, −5.68726015348080852030491991094, −4.86195844916124001924376934642, −4.09438253763086480611998291391, −3.28496892622403112781161009206, −1.99535230939351958695186491382, −0.944935609238651640242504004764,
0.12349593409430536674103668999, 1.68336763172446795525713475775, 3.01061791889135515299391526038, 3.40917700236192021167650707484, 4.07398674660364021702249939811, 5.83620754491163499752752640768, 6.21249697968411824188446421952, 6.56033721953632524892977379361, 7.62236787672623988802049133261, 8.388282488135446445913995311786
