| L(s) = 1 | − 25·5-s + 154.·7-s − 254.·11-s + 928.·13-s − 1.07e3·17-s − 1.12e3·19-s − 3.90e3·23-s + 625·25-s + 211.·29-s + 5.51e3·31-s − 3.86e3·35-s − 1.17e4·37-s − 1.08e4·41-s + 1.19e4·43-s + 2.58e4·47-s + 7.12e3·49-s + 2.20e4·53-s + 6.36e3·55-s − 3.36e4·59-s + 4.01e3·61-s − 2.32e4·65-s − 2.04e4·67-s + 7.55e3·71-s − 7.61e4·73-s − 3.94e4·77-s + 7.93e4·79-s − 1.81e4·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.19·7-s − 0.634·11-s + 1.52·13-s − 0.900·17-s − 0.713·19-s − 1.54·23-s + 0.200·25-s + 0.0466·29-s + 1.03·31-s − 0.533·35-s − 1.40·37-s − 1.00·41-s + 0.987·43-s + 1.70·47-s + 0.423·49-s + 1.07·53-s + 0.283·55-s − 1.25·59-s + 0.138·61-s − 0.681·65-s − 0.555·67-s + 0.177·71-s − 1.67·73-s − 0.757·77-s + 1.43·79-s − 0.289·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 25T \) |
| good | 7 | \( 1 - 154.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 254.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 928.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.07e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.12e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.90e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 211.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.51e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.17e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.08e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.19e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.58e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.20e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.36e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.01e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.04e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 7.55e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.61e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.93e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.81e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.70e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.57e5T + 8.58e9T^{2} \) |
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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
−8.890916420467754209973365992533, −8.373520423928291810274042775279, −7.69353438965169057101861409057, −6.54545505527264294515936238169, −5.62151867247972986193748776591, −4.52653552869781314567331192283, −3.82825856920068013286456852355, −2.37756711303194652295179880894, −1.35492842188018016941346840390, 0,
1.35492842188018016941346840390, 2.37756711303194652295179880894, 3.82825856920068013286456852355, 4.52653552869781314567331192283, 5.62151867247972986193748776591, 6.54545505527264294515936238169, 7.69353438965169057101861409057, 8.373520423928291810274042775279, 8.890916420467754209973365992533
