| L(s) = 1 | − 4·3-s + 25·5-s + 49·7-s − 227·9-s − 124·11-s + 766·13-s − 100·15-s − 1.10e3·17-s + 764·19-s − 196·21-s − 168·23-s + 625·25-s + 1.88e3·27-s − 6.86e3·29-s + 4.09e3·31-s + 496·33-s + 1.22e3·35-s − 4.68e3·37-s − 3.06e3·39-s + 1.31e4·41-s − 1.82e4·43-s − 5.67e3·45-s + 7.10e3·47-s + 2.40e3·49-s + 4.40e3·51-s − 2.00e4·53-s − 3.10e3·55-s + ⋯ |
| L(s) = 1 | − 0.256·3-s + 0.447·5-s + 0.377·7-s − 0.934·9-s − 0.308·11-s + 1.25·13-s − 0.114·15-s − 0.924·17-s + 0.485·19-s − 0.0969·21-s − 0.0662·23-s + 1/5·25-s + 0.496·27-s − 1.51·29-s + 0.765·31-s + 0.0792·33-s + 0.169·35-s − 0.562·37-s − 0.322·39-s + 1.21·41-s − 1.50·43-s − 0.417·45-s + 0.469·47-s + 1/7·49-s + 0.237·51-s − 0.979·53-s − 0.138·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - p^{2} T \) |
| 7 | \( 1 - p^{2} T \) |
| good | 3 | \( 1 + 4 T + p^{5} T^{2} \) |
| 11 | \( 1 + 124 T + p^{5} T^{2} \) |
| 13 | \( 1 - 766 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1102 T + p^{5} T^{2} \) |
| 19 | \( 1 - 764 T + p^{5} T^{2} \) |
| 23 | \( 1 + 168 T + p^{5} T^{2} \) |
| 29 | \( 1 + 6866 T + p^{5} T^{2} \) |
| 31 | \( 1 - 4096 T + p^{5} T^{2} \) |
| 37 | \( 1 + 4682 T + p^{5} T^{2} \) |
| 41 | \( 1 - 13130 T + p^{5} T^{2} \) |
| 43 | \( 1 + 18220 T + p^{5} T^{2} \) |
| 47 | \( 1 - 7104 T + p^{5} T^{2} \) |
| 53 | \( 1 + 20026 T + p^{5} T^{2} \) |
| 59 | \( 1 - 38964 T + p^{5} T^{2} \) |
| 61 | \( 1 + 56274 T + p^{5} T^{2} \) |
| 67 | \( 1 - 24060 T + p^{5} T^{2} \) |
| 71 | \( 1 - 31896 T + p^{5} T^{2} \) |
| 73 | \( 1 + 23670 T + p^{5} T^{2} \) |
| 79 | \( 1 + 37744 T + p^{5} T^{2} \) |
| 83 | \( 1 - 68204 T + p^{5} T^{2} \) |
| 89 | \( 1 + 19078 T + p^{5} T^{2} \) |
| 97 | \( 1 + 115646 T + p^{5} T^{2} \) |
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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
−9.438318291404763985880890745524, −8.685152669227838655495391305481, −7.88587388646985275288181330016, −6.63222832753871593303025555958, −5.83785108991056046048699788075, −5.03981570910434393938250187833, −3.76199176729232888671728479463, −2.56976252640991571393379264775, −1.37297021686375296360078399072, 0,
1.37297021686375296360078399072, 2.56976252640991571393379264775, 3.76199176729232888671728479463, 5.03981570910434393938250187833, 5.83785108991056046048699788075, 6.63222832753871593303025555958, 7.88587388646985275288181330016, 8.685152669227838655495391305481, 9.438318291404763985880890745524
