| L(s) = 1 | + 3-s − 0.732·5-s + 3.46·7-s + 9-s − 11-s + 13-s − 0.732·15-s − 2.73·17-s + 2·19-s + 3.46·21-s − 6·23-s − 4.46·25-s + 27-s − 6.73·29-s − 5.26·31-s − 33-s − 2.53·35-s − 2·37-s + 39-s − 10.9·41-s − 3.26·43-s − 0.732·45-s + 1.46·47-s + 4.99·49-s − 2.73·51-s − 6·53-s + 0.732·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.327·5-s + 1.30·7-s + 0.333·9-s − 0.301·11-s + 0.277·13-s − 0.189·15-s − 0.662·17-s + 0.458·19-s + 0.755·21-s − 1.25·23-s − 0.892·25-s + 0.192·27-s − 1.25·29-s − 0.946·31-s − 0.174·33-s − 0.428·35-s − 0.328·37-s + 0.160·39-s − 1.70·41-s − 0.498·43-s − 0.109·45-s + 0.213·47-s + 0.714·49-s − 0.382·51-s − 0.824·53-s + 0.0987·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 0.732T + 5T^{2} \) |
| 7 | \( 1 - 3.46T + 7T^{2} \) |
| 17 | \( 1 + 2.73T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 + 6.73T + 29T^{2} \) |
| 31 | \( 1 + 5.26T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 10.9T + 41T^{2} \) |
| 43 | \( 1 + 3.26T + 43T^{2} \) |
| 47 | \( 1 - 1.46T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 5.46T + 59T^{2} \) |
| 61 | \( 1 - 4.92T + 61T^{2} \) |
| 67 | \( 1 + 8.19T + 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 - 3.26T + 79T^{2} \) |
| 83 | \( 1 + 10.9T + 83T^{2} \) |
| 89 | \( 1 - 11.6T + 89T^{2} \) |
| 97 | \( 1 - 15.8T + 97T^{2} \) |
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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
−7.69597131791864015009362641767, −7.22241049868047777774886470892, −6.17840073585047549482521104897, −5.39629118391642939488247088111, −4.69026469930325998227201605605, −3.94713003019103775969752572938, −3.28278953456966509281594157748, −2.02229211029712255447480411190, −1.65962894036552175605388417816, 0,
1.65962894036552175605388417816, 2.02229211029712255447480411190, 3.28278953456966509281594157748, 3.94713003019103775969752572938, 4.69026469930325998227201605605, 5.39629118391642939488247088111, 6.17840073585047549482521104897, 7.22241049868047777774886470892, 7.69597131791864015009362641767
