| L(s) = 1 | + 3·3-s + 7·5-s + 9·9-s − 7·11-s − 52·13-s + 21·15-s + 72·17-s − 20·19-s + 48·23-s − 76·25-s + 27·27-s − 243·29-s − 95·31-s − 21·33-s + 352·37-s − 156·39-s − 296·41-s − 158·43-s + 63·45-s + 142·47-s + 216·51-s − 375·53-s − 49·55-s − 60·57-s − 279·59-s + 246·61-s − 364·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.626·5-s + 1/3·9-s − 0.191·11-s − 1.10·13-s + 0.361·15-s + 1.02·17-s − 0.241·19-s + 0.435·23-s − 0.607·25-s + 0.192·27-s − 1.55·29-s − 0.550·31-s − 0.110·33-s + 1.56·37-s − 0.640·39-s − 1.12·41-s − 0.560·43-s + 0.208·45-s + 0.440·47-s + 0.593·51-s − 0.971·53-s − 0.120·55-s − 0.139·57-s − 0.615·59-s + 0.516·61-s − 0.694·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 7 T + p^{3} T^{2} \) |
| 11 | \( 1 + 7 T + p^{3} T^{2} \) |
| 13 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 17 | \( 1 - 72 T + p^{3} T^{2} \) |
| 19 | \( 1 + 20 T + p^{3} T^{2} \) |
| 23 | \( 1 - 48 T + p^{3} T^{2} \) |
| 29 | \( 1 + 243 T + p^{3} T^{2} \) |
| 31 | \( 1 + 95 T + p^{3} T^{2} \) |
| 37 | \( 1 - 352 T + p^{3} T^{2} \) |
| 41 | \( 1 + 296 T + p^{3} T^{2} \) |
| 43 | \( 1 + 158 T + p^{3} T^{2} \) |
| 47 | \( 1 - 142 T + p^{3} T^{2} \) |
| 53 | \( 1 + 375 T + p^{3} T^{2} \) |
| 59 | \( 1 + 279 T + p^{3} T^{2} \) |
| 61 | \( 1 - 246 T + p^{3} T^{2} \) |
| 67 | \( 1 - 730 T + p^{3} T^{2} \) |
| 71 | \( 1 + 338 T + p^{3} T^{2} \) |
| 73 | \( 1 + 542 T + p^{3} T^{2} \) |
| 79 | \( 1 - 305 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1123 T + p^{3} T^{2} \) |
| 89 | \( 1 + 426 T + p^{3} T^{2} \) |
| 97 | \( 1 + 369 T + p^{3} T^{2} \) |
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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
−8.140132737905911817301461753372, −7.58515541562125544914386312208, −6.82907099889987585487124390835, −5.77867804296397847211493496107, −5.17875015434317452774319253715, −4.14763924057246665796787717606, −3.17729147880147142547810934568, −2.31640090420773729203201243034, −1.45844729916602329921942355112, 0,
1.45844729916602329921942355112, 2.31640090420773729203201243034, 3.17729147880147142547810934568, 4.14763924057246665796787717606, 5.17875015434317452774319253715, 5.77867804296397847211493496107, 6.82907099889987585487124390835, 7.58515541562125544914386312208, 8.140132737905911817301461753372
