| L(s) = 1 | + 3.08·3-s + 9.08·5-s − 7·7-s − 17.4·9-s − 34.1·11-s − 13.0·13-s + 28.0·15-s − 12.8·17-s − 139.·19-s − 21.5·21-s − 21.3·23-s − 42.5·25-s − 137.·27-s − 142.·29-s + 192.·31-s − 105.·33-s − 63.5·35-s + 115.·37-s − 40.3·39-s + 396.·41-s − 252.·43-s − 158.·45-s + 11.8·47-s + 49·49-s − 39.5·51-s + 295.·53-s − 310.·55-s + ⋯ |
| L(s) = 1 | + 0.593·3-s + 0.812·5-s − 0.377·7-s − 0.648·9-s − 0.936·11-s − 0.279·13-s + 0.481·15-s − 0.183·17-s − 1.68·19-s − 0.224·21-s − 0.193·23-s − 0.340·25-s − 0.977·27-s − 0.914·29-s + 1.11·31-s − 0.555·33-s − 0.307·35-s + 0.514·37-s − 0.165·39-s + 1.51·41-s − 0.895·43-s − 0.526·45-s + 0.0366·47-s + 0.142·49-s − 0.108·51-s + 0.765·53-s − 0.760·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 3 | \( 1 - 3.08T + 27T^{2} \) |
| 5 | \( 1 - 9.08T + 125T^{2} \) |
| 11 | \( 1 + 34.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 13.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 12.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 139.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 21.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 142.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 192.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 115.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 396.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 252.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 11.8T + 1.03e5T^{2} \) |
| 53 | \( 1 - 295.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 269.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 233.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 975.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 870.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 391.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 897.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 658.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 32.6T + 7.04e5T^{2} \) |
| 97 | \( 1 - 991.T + 9.12e5T^{2} \) |
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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
−10.15612018421789160141436198851, −9.311802383005036518379424147644, −8.486016419299810649323957656969, −7.62295204434064096934924916692, −6.31775688468060558928645342428, −5.61115122192604881095086358673, −4.28280949735376697737110068607, −2.84969665589894616506925621521, −2.09232312499618917596381494741, 0,
2.09232312499618917596381494741, 2.84969665589894616506925621521, 4.28280949735376697737110068607, 5.61115122192604881095086358673, 6.31775688468060558928645342428, 7.62295204434064096934924916692, 8.486016419299810649323957656969, 9.311802383005036518379424147644, 10.15612018421789160141436198851
