| L(s) = 1 | + 0.614·3-s + 5-s − 2.24·7-s − 2.62·9-s + 5.19·13-s + 0.614·15-s − 3.22·17-s − 7.19·19-s − 1.38·21-s + 5.11·23-s + 25-s − 3.45·27-s + 9.61·29-s + 0.491·31-s − 2.24·35-s + 0.350·37-s + 3.19·39-s − 4.42·41-s − 10.9·43-s − 2.62·45-s − 1.03·47-s − 1.95·49-s − 1.98·51-s + 3.75·53-s − 4.42·57-s + 12.7·59-s − 6.38·61-s + ⋯ |
| L(s) = 1 | + 0.355·3-s + 0.447·5-s − 0.848·7-s − 0.873·9-s + 1.44·13-s + 0.158·15-s − 0.783·17-s − 1.65·19-s − 0.301·21-s + 1.06·23-s + 0.200·25-s − 0.665·27-s + 1.78·29-s + 0.0882·31-s − 0.379·35-s + 0.0576·37-s + 0.511·39-s − 0.690·41-s − 1.67·43-s − 0.390·45-s − 0.150·47-s − 0.279·49-s − 0.278·51-s + 0.515·53-s − 0.586·57-s + 1.65·59-s − 0.818·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 0.614T + 3T^{2} \) |
| 7 | \( 1 + 2.24T + 7T^{2} \) |
| 13 | \( 1 - 5.19T + 13T^{2} \) |
| 17 | \( 1 + 3.22T + 17T^{2} \) |
| 19 | \( 1 + 7.19T + 19T^{2} \) |
| 23 | \( 1 - 5.11T + 23T^{2} \) |
| 29 | \( 1 - 9.61T + 29T^{2} \) |
| 31 | \( 1 - 0.491T + 31T^{2} \) |
| 37 | \( 1 - 0.350T + 37T^{2} \) |
| 41 | \( 1 + 4.42T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 + 1.03T + 47T^{2} \) |
| 53 | \( 1 - 3.75T + 53T^{2} \) |
| 59 | \( 1 - 12.7T + 59T^{2} \) |
| 61 | \( 1 + 6.38T + 61T^{2} \) |
| 67 | \( 1 + 8.44T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 - 2.13T + 79T^{2} \) |
| 83 | \( 1 + 10.7T + 83T^{2} \) |
| 89 | \( 1 - 9.90T + 89T^{2} \) |
| 97 | \( 1 + 15.9T + 97T^{2} \) |
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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
−8.290423004444549360962734998004, −6.93411371138750262170456244985, −6.42987597082227600940222572622, −5.97200125207206070418377572673, −4.95680075765627903543513527458, −4.06553203345657894910602162357, −3.16405449445578732334569760020, −2.59117423742805720460575045348, −1.44676036649645738241952603334, 0,
1.44676036649645738241952603334, 2.59117423742805720460575045348, 3.16405449445578732334569760020, 4.06553203345657894910602162357, 4.95680075765627903543513527458, 5.97200125207206070418377572673, 6.42987597082227600940222572622, 6.93411371138750262170456244985, 8.290423004444549360962734998004
