| L(s) = 1 | + 0.732·3-s + 1.73·5-s − 7-s − 2.46·9-s − 4.73·11-s + 13-s + 1.26·15-s − 2.19·17-s − 7.19·19-s − 0.732·21-s − 3·23-s − 2.00·25-s − 4·27-s + 0.464·29-s − 1.19·31-s − 3.46·33-s − 1.73·35-s + 4.19·37-s + 0.732·39-s + 3.46·41-s + 7·43-s − 4.26·45-s − 7.73·47-s + 49-s − 1.60·51-s − 9.92·53-s − 8.19·55-s + ⋯ |
| L(s) = 1 | + 0.422·3-s + 0.774·5-s − 0.377·7-s − 0.821·9-s − 1.42·11-s + 0.277·13-s + 0.327·15-s − 0.532·17-s − 1.65·19-s − 0.159·21-s − 0.625·23-s − 0.400·25-s − 0.769·27-s + 0.0861·29-s − 0.214·31-s − 0.603·33-s − 0.292·35-s + 0.689·37-s + 0.117·39-s + 0.541·41-s + 1.06·43-s − 0.636·45-s − 1.12·47-s + 0.142·49-s − 0.225·51-s − 1.36·53-s − 1.10·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 0.732T + 3T^{2} \) |
| 5 | \( 1 - 1.73T + 5T^{2} \) |
| 11 | \( 1 + 4.73T + 11T^{2} \) |
| 17 | \( 1 + 2.19T + 17T^{2} \) |
| 19 | \( 1 + 7.19T + 19T^{2} \) |
| 23 | \( 1 + 3T + 23T^{2} \) |
| 29 | \( 1 - 0.464T + 29T^{2} \) |
| 31 | \( 1 + 1.19T + 31T^{2} \) |
| 37 | \( 1 - 4.19T + 37T^{2} \) |
| 41 | \( 1 - 3.46T + 41T^{2} \) |
| 43 | \( 1 - 7T + 43T^{2} \) |
| 47 | \( 1 + 7.73T + 47T^{2} \) |
| 53 | \( 1 + 9.92T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 14.3T + 67T^{2} \) |
| 71 | \( 1 - 1.26T + 71T^{2} \) |
| 73 | \( 1 + 9.19T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 - 0.803T + 83T^{2} \) |
| 89 | \( 1 - 11.1T + 89T^{2} \) |
| 97 | \( 1 - 2.80T + 97T^{2} \) |
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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
−9.132736920368041159996276173499, −8.299533040819314584641077226243, −7.75321836221106315023913677734, −6.42686067911709445685198931107, −5.94646945513252609890209384589, −5.01291835484182361803191855382, −3.87378786348003235834413755376, −2.67002418363340391217245064633, −2.11279221226059992211318513525, 0,
2.11279221226059992211318513525, 2.67002418363340391217245064633, 3.87378786348003235834413755376, 5.01291835484182361803191855382, 5.94646945513252609890209384589, 6.42686067911709445685198931107, 7.75321836221106315023913677734, 8.299533040819314584641077226243, 9.132736920368041159996276173499
