L(s) = 1 | − 4·2-s + 10.7·3-s + 16·4-s + 59.4·5-s − 42.8·6-s − 230.·7-s − 64·8-s − 128.·9-s − 237.·10-s − 57.4·11-s + 171.·12-s + 1.07e3·13-s + 921.·14-s + 636.·15-s + 256·16-s − 863.·17-s + 513.·18-s + 950.·20-s − 2.46e3·21-s + 229.·22-s + 1.59e3·23-s − 685.·24-s + 404.·25-s − 4.31e3·26-s − 3.97e3·27-s − 3.68e3·28-s + 7.23e3·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.687·3-s + 0.5·4-s + 1.06·5-s − 0.485·6-s − 1.77·7-s − 0.353·8-s − 0.527·9-s − 0.751·10-s − 0.143·11-s + 0.343·12-s + 1.76·13-s + 1.25·14-s + 0.730·15-s + 0.250·16-s − 0.724·17-s + 0.373·18-s + 0.531·20-s − 1.22·21-s + 0.101·22-s + 0.628·23-s − 0.242·24-s + 0.129·25-s − 1.25·26-s − 1.04·27-s − 0.888·28-s + 1.59·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.765800814\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.765800814\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 4T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 - 10.7T + 243T^{2} \) |
| 5 | \( 1 - 59.4T + 3.12e3T^{2} \) |
| 7 | \( 1 + 230.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 57.4T + 1.61e5T^{2} \) |
| 13 | \( 1 - 1.07e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 863.T + 1.41e6T^{2} \) |
| 23 | \( 1 - 1.59e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.23e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.20e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.44e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.46e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.73e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 4.52e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 610.T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.08e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.70e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 7.49e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.95e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.23e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.08e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.74e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.35e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 4.20e4T + 8.58e9T^{2} \) |
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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
−9.462147109888359292405568975232, −8.897302387815469965723065227362, −8.329697738216647056544747772714, −6.82189493221643745869477821599, −6.35642639508262881991311220412, −5.53736115106861198957810468030, −3.64014045215299594129479550744, −2.95510552474010287206240935743, −1.98084438198127367612265845884, −0.64634222763304530955124053936,
0.64634222763304530955124053936, 1.98084438198127367612265845884, 2.95510552474010287206240935743, 3.64014045215299594129479550744, 5.53736115106861198957810468030, 6.35642639508262881991311220412, 6.82189493221643745869477821599, 8.329697738216647056544747772714, 8.897302387815469965723065227362, 9.462147109888359292405568975232