| L(s) = 1 | − 2.02·3-s + 2.88·5-s + 7-s + 1.11·9-s + 11-s + 13-s − 5.85·15-s + 4.54·17-s + 4.10·19-s − 2.02·21-s − 7.64·23-s + 3.34·25-s + 3.82·27-s − 6.29·29-s − 9.32·31-s − 2.02·33-s + 2.88·35-s − 3.59·37-s − 2.02·39-s + 2.53·41-s − 9.96·43-s + 3.21·45-s − 0.779·47-s + 49-s − 9.21·51-s − 13.7·53-s + 2.88·55-s + ⋯ |
| L(s) = 1 | − 1.17·3-s + 1.29·5-s + 0.377·7-s + 0.370·9-s + 0.301·11-s + 0.277·13-s − 1.51·15-s + 1.10·17-s + 0.940·19-s − 0.442·21-s − 1.59·23-s + 0.669·25-s + 0.736·27-s − 1.16·29-s − 1.67·31-s − 0.352·33-s + 0.488·35-s − 0.590·37-s − 0.324·39-s + 0.396·41-s − 1.52·43-s + 0.478·45-s − 0.113·47-s + 0.142·49-s − 1.28·51-s − 1.88·53-s + 0.389·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 2.02T + 3T^{2} \) |
| 5 | \( 1 - 2.88T + 5T^{2} \) |
| 17 | \( 1 - 4.54T + 17T^{2} \) |
| 19 | \( 1 - 4.10T + 19T^{2} \) |
| 23 | \( 1 + 7.64T + 23T^{2} \) |
| 29 | \( 1 + 6.29T + 29T^{2} \) |
| 31 | \( 1 + 9.32T + 31T^{2} \) |
| 37 | \( 1 + 3.59T + 37T^{2} \) |
| 41 | \( 1 - 2.53T + 41T^{2} \) |
| 43 | \( 1 + 9.96T + 43T^{2} \) |
| 47 | \( 1 + 0.779T + 47T^{2} \) |
| 53 | \( 1 + 13.7T + 53T^{2} \) |
| 59 | \( 1 - 0.314T + 59T^{2} \) |
| 61 | \( 1 - 5.24T + 61T^{2} \) |
| 67 | \( 1 + 14.2T + 67T^{2} \) |
| 71 | \( 1 + 2.27T + 71T^{2} \) |
| 73 | \( 1 - 7.19T + 73T^{2} \) |
| 79 | \( 1 + 2.06T + 79T^{2} \) |
| 83 | \( 1 + 0.240T + 83T^{2} \) |
| 89 | \( 1 + 10.2T + 89T^{2} \) |
| 97 | \( 1 + 5.32T + 97T^{2} \) |
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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
−7.41033224411129836846465158186, −6.53323803090705874606226564815, −5.94270196516651126302533437779, −5.47750616577619458664764171217, −5.09960987429817320164237064647, −3.96830778125452706987745612452, −3.13237859801300528386401488690, −1.85662627675763665583787457556, −1.39401451660083662522470921493, 0,
1.39401451660083662522470921493, 1.85662627675763665583787457556, 3.13237859801300528386401488690, 3.96830778125452706987745612452, 5.09960987429817320164237064647, 5.47750616577619458664764171217, 5.94270196516651126302533437779, 6.53323803090705874606226564815, 7.41033224411129836846465158186
