L(s) = 1 | + 8.44·2-s + 39.3·4-s − 36·5-s + 61.9·8-s − 304.·10-s − 295.·11-s + 1.14e3·13-s − 735.·16-s + 1.03e3·17-s − 2.10e3·19-s − 1.41e3·20-s − 2.49e3·22-s + 640.·23-s − 1.82e3·25-s + 9.69e3·26-s − 7.63e3·29-s + 966.·31-s − 8.19e3·32-s + 8.71e3·34-s − 1.77e3·37-s − 1.78e4·38-s − 2.23e3·40-s − 1.19e4·41-s − 1.98e4·43-s − 1.16e4·44-s + 5.41e3·46-s − 2.79e4·47-s + ⋯ |
L(s) = 1 | + 1.49·2-s + 1.22·4-s − 0.643·5-s + 0.342·8-s − 0.961·10-s − 0.736·11-s + 1.88·13-s − 0.718·16-s + 0.866·17-s − 1.33·19-s − 0.791·20-s − 1.09·22-s + 0.252·23-s − 0.585·25-s + 2.81·26-s − 1.68·29-s + 0.180·31-s − 1.41·32-s + 1.29·34-s − 0.212·37-s − 2.00·38-s − 0.220·40-s − 1.11·41-s − 1.63·43-s − 0.905·44-s + 0.377·46-s − 1.84·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 8.44T + 32T^{2} \) |
| 5 | \( 1 + 36T + 3.12e3T^{2} \) |
| 11 | \( 1 + 295.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 1.14e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.03e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.10e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 640.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.63e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 966.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.77e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.19e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.98e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.79e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 7.11e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.08e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.38e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.46e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.84e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.52e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.30e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.03e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.60e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.53e5T + 8.58e9T^{2} \) |
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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
−10.16796018554225294144428298000, −8.720666654300518761836006854296, −7.958810638465216044128767776331, −6.70410558402122803501774336586, −5.87979325166003528392137598957, −4.98152861759306868455910770502, −3.81958507626244868794461265985, −3.34968875311475324579506409503, −1.81855741798239935593978434442, 0,
1.81855741798239935593978434442, 3.34968875311475324579506409503, 3.81958507626244868794461265985, 4.98152861759306868455910770502, 5.87979325166003528392137598957, 6.70410558402122803501774336586, 7.958810638465216044128767776331, 8.720666654300518761836006854296, 10.16796018554225294144428298000