| L(s) = 1 | + 3·3-s − 2·5-s + 9·9-s − 18·11-s − 33·13-s − 6·15-s + 68·17-s − 25·19-s + 92·23-s − 121·25-s + 27·27-s + 92·29-s − 25·31-s − 54·33-s − 213·37-s − 99·39-s − 94·41-s − 67·43-s − 18·45-s − 278·47-s + 204·51-s − 400·53-s + 36·55-s − 75·57-s − 744·59-s + 734·61-s + 66·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.178·5-s + 1/3·9-s − 0.493·11-s − 0.704·13-s − 0.103·15-s + 0.970·17-s − 0.301·19-s + 0.834·23-s − 0.967·25-s + 0.192·27-s + 0.589·29-s − 0.144·31-s − 0.284·33-s − 0.946·37-s − 0.406·39-s − 0.358·41-s − 0.237·43-s − 0.0596·45-s − 0.862·47-s + 0.560·51-s − 1.03·53-s + 0.0882·55-s − 0.174·57-s − 1.64·59-s + 1.54·61-s + 0.125·65-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 | \( 1 + 2 T + p^{3} T^{2} \) |
| 11 | \( 1 + 18 T + p^{3} T^{2} \) |
| 13 | \( 1 + 33 T + p^{3} T^{2} \) |
| 17 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 19 | \( 1 + 25 T + p^{3} T^{2} \) |
| 23 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 29 | \( 1 - 92 T + p^{3} T^{2} \) |
| 31 | \( 1 + 25 T + p^{3} T^{2} \) |
| 37 | \( 1 + 213 T + p^{3} T^{2} \) |
| 41 | \( 1 + 94 T + p^{3} T^{2} \) |
| 43 | \( 1 + 67 T + p^{3} T^{2} \) |
| 47 | \( 1 + 278 T + p^{3} T^{2} \) |
| 53 | \( 1 + 400 T + p^{3} T^{2} \) |
| 59 | \( 1 + 744 T + p^{3} T^{2} \) |
| 61 | \( 1 - 734 T + p^{3} T^{2} \) |
| 67 | \( 1 - 555 T + p^{3} T^{2} \) |
| 71 | \( 1 + 642 T + p^{3} T^{2} \) |
| 73 | \( 1 + 973 T + p^{3} T^{2} \) |
| 79 | \( 1 + 785 T + p^{3} T^{2} \) |
| 83 | \( 1 - 822 T + p^{3} T^{2} \) |
| 89 | \( 1 + 424 T + p^{3} T^{2} \) |
| 97 | \( 1 - 734 T + p^{3} T^{2} \) |
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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
−8.956041609229332078267327042733, −8.099486303909547350816176462221, −7.50743901221251428923618930010, −6.62597436586790268154762447995, −5.46989023392672659452710904146, −4.65894357033359863210984318101, −3.52991103327414288379386211400, −2.70111781575931940553113333781, −1.51674196883343001704596656162, 0,
1.51674196883343001704596656162, 2.70111781575931940553113333781, 3.52991103327414288379386211400, 4.65894357033359863210984318101, 5.46989023392672659452710904146, 6.62597436586790268154762447995, 7.50743901221251428923618930010, 8.099486303909547350816176462221, 8.956041609229332078267327042733
