| L(s) = 1 | − 3·3-s − 10·5-s + 7·7-s + 9·9-s + 52·11-s − 10·13-s + 30·15-s − 54·17-s + 52·19-s − 21·21-s − 48·23-s − 25·25-s − 27·27-s − 186·29-s − 224·31-s − 156·33-s − 70·35-s + 94·37-s + 30·39-s − 478·41-s + 316·43-s − 90·45-s − 256·47-s + 49·49-s + 162·51-s − 66·53-s − 520·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s + 1.42·11-s − 0.213·13-s + 0.516·15-s − 0.770·17-s + 0.627·19-s − 0.218·21-s − 0.435·23-s − 1/5·25-s − 0.192·27-s − 1.19·29-s − 1.29·31-s − 0.822·33-s − 0.338·35-s + 0.417·37-s + 0.123·39-s − 1.82·41-s + 1.12·43-s − 0.298·45-s − 0.794·47-s + 1/7·49-s + 0.444·51-s − 0.171·53-s − 1.27·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p T \) |
| good | 5 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 11 | \( 1 - 52 T + p^{3} T^{2} \) |
| 13 | \( 1 + 10 T + p^{3} T^{2} \) |
| 17 | \( 1 + 54 T + p^{3} T^{2} \) |
| 19 | \( 1 - 52 T + p^{3} T^{2} \) |
| 23 | \( 1 + 48 T + p^{3} T^{2} \) |
| 29 | \( 1 + 186 T + p^{3} T^{2} \) |
| 31 | \( 1 + 224 T + p^{3} T^{2} \) |
| 37 | \( 1 - 94 T + p^{3} T^{2} \) |
| 41 | \( 1 + 478 T + p^{3} T^{2} \) |
| 43 | \( 1 - 316 T + p^{3} T^{2} \) |
| 47 | \( 1 + 256 T + p^{3} T^{2} \) |
| 53 | \( 1 + 66 T + p^{3} T^{2} \) |
| 59 | \( 1 + 420 T + p^{3} T^{2} \) |
| 61 | \( 1 - 342 T + p^{3} T^{2} \) |
| 67 | \( 1 + 668 T + p^{3} T^{2} \) |
| 71 | \( 1 - 272 T + p^{3} T^{2} \) |
| 73 | \( 1 + 86 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1360 T + p^{3} T^{2} \) |
| 83 | \( 1 + 188 T + p^{3} T^{2} \) |
| 89 | \( 1 + 366 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1554 T + p^{3} T^{2} \) |
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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
−11.00438315739502686108727715319, −9.705202424256222519039528277498, −8.818123833378362311540019908528, −7.65784883778203192769729337697, −6.86730016472715386841331681963, −5.72119366176637657418582837601, −4.46893636727600929425998276421, −3.63226233488322435021179897075, −1.62871813279808016586018622682, 0,
1.62871813279808016586018622682, 3.63226233488322435021179897075, 4.46893636727600929425998276421, 5.72119366176637657418582837601, 6.86730016472715386841331681963, 7.65784883778203192769729337697, 8.818123833378362311540019908528, 9.705202424256222519039528277498, 11.00438315739502686108727715319
