| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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
−9.320532255918726978174241247199, −7.972782200041578910782554404917, −7.36728424810653286340310842364, −6.31049702346212859083536453987, −5.62696294655573401773815686411, −4.85440621183443279098503602696, −3.74272478329037057404671688353, −2.47450270127512902018460749200, −1.37792169243907164873835146151, 0,
1.37792169243907164873835146151, 2.47450270127512902018460749200, 3.74272478329037057404671688353, 4.85440621183443279098503602696, 5.62696294655573401773815686411, 6.31049702346212859083536453987, 7.36728424810653286340310842364, 7.972782200041578910782554404917, 9.320532255918726978174241247199
