| L(s) = 1 | + 3-s − 3.82·5-s + 3.22·7-s + 9-s + 3.21·11-s − 3.97·13-s − 3.82·15-s − 2.27·17-s + 0.0375·19-s + 3.22·21-s + 1.29·23-s + 9.66·25-s + 27-s − 4.68·29-s − 4.65·31-s + 3.21·33-s − 12.3·35-s − 2.10·37-s − 3.97·39-s − 9.37·41-s + 10.1·43-s − 3.82·45-s − 9.03·47-s + 3.37·49-s − 2.27·51-s − 0.602·53-s − 12.3·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.71·5-s + 1.21·7-s + 0.333·9-s + 0.970·11-s − 1.10·13-s − 0.988·15-s − 0.552·17-s + 0.00860·19-s + 0.702·21-s + 0.269·23-s + 1.93·25-s + 0.192·27-s − 0.870·29-s − 0.835·31-s + 0.560·33-s − 2.08·35-s − 0.346·37-s − 0.636·39-s − 1.46·41-s + 1.55·43-s − 0.570·45-s − 1.31·47-s + 0.481·49-s − 0.319·51-s − 0.0828·53-s − 1.66·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 503 | \( 1 + T \) |
| good | 5 | \( 1 + 3.82T + 5T^{2} \) |
| 7 | \( 1 - 3.22T + 7T^{2} \) |
| 11 | \( 1 - 3.21T + 11T^{2} \) |
| 13 | \( 1 + 3.97T + 13T^{2} \) |
| 17 | \( 1 + 2.27T + 17T^{2} \) |
| 19 | \( 1 - 0.0375T + 19T^{2} \) |
| 23 | \( 1 - 1.29T + 23T^{2} \) |
| 29 | \( 1 + 4.68T + 29T^{2} \) |
| 31 | \( 1 + 4.65T + 31T^{2} \) |
| 37 | \( 1 + 2.10T + 37T^{2} \) |
| 41 | \( 1 + 9.37T + 41T^{2} \) |
| 43 | \( 1 - 10.1T + 43T^{2} \) |
| 47 | \( 1 + 9.03T + 47T^{2} \) |
| 53 | \( 1 + 0.602T + 53T^{2} \) |
| 59 | \( 1 - 3.58T + 59T^{2} \) |
| 61 | \( 1 + 3.24T + 61T^{2} \) |
| 67 | \( 1 - 12.9T + 67T^{2} \) |
| 71 | \( 1 + 5.99T + 71T^{2} \) |
| 73 | \( 1 - 0.896T + 73T^{2} \) |
| 79 | \( 1 - 2.02T + 79T^{2} \) |
| 83 | \( 1 - 0.335T + 83T^{2} \) |
| 89 | \( 1 - 6.96T + 89T^{2} \) |
| 97 | \( 1 + 5.63T + 97T^{2} \) |
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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
−7.75834344945657938596452010568, −7.23750077875944219479252611582, −6.65660218984571467075188880912, −5.25667550290995654853918898530, −4.67262333237170177870799542033, −4.00378844430792608323824922902, −3.44260089303418382175266424552, −2.32724918308872616838923837090, −1.36566509646770508664526433346, 0,
1.36566509646770508664526433346, 2.32724918308872616838923837090, 3.44260089303418382175266424552, 4.00378844430792608323824922902, 4.67262333237170177870799542033, 5.25667550290995654853918898530, 6.65660218984571467075188880912, 7.23750077875944219479252611582, 7.75834344945657938596452010568
