| L(s) = 1 | + 2.56·3-s − 2.56·7-s + 3.56·9-s + 5.12·11-s − 13-s − 5.68·17-s − 5.12·19-s − 6.56·21-s − 8·23-s + 1.43·27-s − 2·29-s − 4·31-s + 13.1·33-s − 9.68·37-s − 2.56·39-s − 3.12·41-s + 5.43·43-s − 0.315·47-s − 0.438·49-s − 14.5·51-s − 3.12·53-s − 13.1·57-s − 5.12·59-s + 11.1·61-s − 9.12·63-s − 5.12·67-s − 20.4·69-s + ⋯ |
| L(s) = 1 | + 1.47·3-s − 0.968·7-s + 1.18·9-s + 1.54·11-s − 0.277·13-s − 1.37·17-s − 1.17·19-s − 1.43·21-s − 1.66·23-s + 0.276·27-s − 0.371·29-s − 0.718·31-s + 2.28·33-s − 1.59·37-s − 0.410·39-s − 0.487·41-s + 0.829·43-s − 0.0459·47-s − 0.0626·49-s − 2.03·51-s − 0.428·53-s − 1.73·57-s − 0.666·59-s + 1.42·61-s − 1.14·63-s − 0.625·67-s − 2.46·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 2.56T + 3T^{2} \) |
| 7 | \( 1 + 2.56T + 7T^{2} \) |
| 11 | \( 1 - 5.12T + 11T^{2} \) |
| 17 | \( 1 + 5.68T + 17T^{2} \) |
| 19 | \( 1 + 5.12T + 19T^{2} \) |
| 23 | \( 1 + 8T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 9.68T + 37T^{2} \) |
| 41 | \( 1 + 3.12T + 41T^{2} \) |
| 43 | \( 1 - 5.43T + 43T^{2} \) |
| 47 | \( 1 + 0.315T + 47T^{2} \) |
| 53 | \( 1 + 3.12T + 53T^{2} \) |
| 59 | \( 1 + 5.12T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 + 5.12T + 67T^{2} \) |
| 71 | \( 1 - 7.68T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 2.24T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 8.24T + 97T^{2} \) |
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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
−8.026642507422669761059204455682, −7.07707774038457988039724322452, −6.59862760049216744189215010458, −5.92981294406266044089713846339, −4.54406800678627966732615865159, −3.83792058552520780745080226009, −3.45526606635599252297821746820, −2.28125565019758375297421050789, −1.82242405707376003780275513056, 0,
1.82242405707376003780275513056, 2.28125565019758375297421050789, 3.45526606635599252297821746820, 3.83792058552520780745080226009, 4.54406800678627966732615865159, 5.92981294406266044089713846339, 6.59862760049216744189215010458, 7.07707774038457988039724322452, 8.026642507422669761059204455682
