| L(s) = 1 | − 0.904·2-s − 9.08·3-s − 7.18·4-s + 3.14·5-s + 8.21·6-s + 13.7·8-s + 55.5·9-s − 2.84·10-s + 53.3·11-s + 65.2·12-s + 57.9·13-s − 28.5·15-s + 45.0·16-s − 76.7·17-s − 50.2·18-s − 20.7·19-s − 22.5·20-s − 48.2·22-s + 66.2·23-s − 124.·24-s − 115.·25-s − 52.4·26-s − 259.·27-s + 27.2·29-s + 25.8·30-s − 294.·31-s − 150.·32-s + ⋯ |
| L(s) = 1 | − 0.319·2-s − 1.74·3-s − 0.897·4-s + 0.280·5-s + 0.559·6-s + 0.606·8-s + 2.05·9-s − 0.0898·10-s + 1.46·11-s + 1.56·12-s + 1.23·13-s − 0.491·15-s + 0.703·16-s − 1.09·17-s − 0.658·18-s − 0.250·19-s − 0.252·20-s − 0.468·22-s + 0.601·23-s − 1.06·24-s − 0.921·25-s − 0.395·26-s − 1.84·27-s + 0.174·29-s + 0.157·30-s − 1.70·31-s − 0.831·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 + 0.904T + 8T^{2} \) |
| 3 | \( 1 + 9.08T + 27T^{2} \) |
| 5 | \( 1 - 3.14T + 125T^{2} \) |
| 11 | \( 1 - 53.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 57.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 76.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 20.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 66.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 27.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 294.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 33.3T + 5.06e4T^{2} \) |
| 41 | \( 1 - 92.5T + 6.89e4T^{2} \) |
| 43 | \( 1 + 321.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 393.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 514.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 113.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 258.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 49.2T + 3.00e5T^{2} \) |
| 71 | \( 1 - 461.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 428.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 93.0T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.15e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 581.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 348.T + 9.12e5T^{2} \) |
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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
−8.429797626378976256330990049689, −7.19482649855224693900598693277, −6.54785568769996642707746442978, −5.86298408622213252653785366405, −5.22440428176980097125839347333, −4.20207192355098703188704188436, −3.84329231853833850066755643772, −1.70248062579647772639319551308, −0.972234617892310666701506799146, 0,
0.972234617892310666701506799146, 1.70248062579647772639319551308, 3.84329231853833850066755643772, 4.20207192355098703188704188436, 5.22440428176980097125839347333, 5.86298408622213252653785366405, 6.54785568769996642707746442978, 7.19482649855224693900598693277, 8.429797626378976256330990049689
