| L(s) = 1 | + 3-s + 0.585·5-s + 9-s − 4.82·11-s + 4.24·13-s + 0.585·15-s − 4.58·17-s − 1.17·19-s − 0.828·23-s − 4.65·25-s + 27-s + 2.82·29-s + 2.82·31-s − 4.82·33-s − 9.65·37-s + 4.24·39-s − 1.75·41-s + 11.3·43-s + 0.585·45-s + 12.4·47-s − 4.58·51-s + 2·53-s − 2.82·55-s − 1.17·57-s + 8.48·59-s − 3.07·61-s + 2.48·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.261·5-s + 0.333·9-s − 1.45·11-s + 1.17·13-s + 0.151·15-s − 1.11·17-s − 0.268·19-s − 0.172·23-s − 0.931·25-s + 0.192·27-s + 0.525·29-s + 0.508·31-s − 0.840·33-s − 1.58·37-s + 0.679·39-s − 0.274·41-s + 1.72·43-s + 0.0873·45-s + 1.82·47-s − 0.642·51-s + 0.274·53-s − 0.381·55-s − 0.155·57-s + 1.10·59-s − 0.393·61-s + 0.308·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 0.585T + 5T^{2} \) |
| 11 | \( 1 + 4.82T + 11T^{2} \) |
| 13 | \( 1 - 4.24T + 13T^{2} \) |
| 17 | \( 1 + 4.58T + 17T^{2} \) |
| 19 | \( 1 + 1.17T + 19T^{2} \) |
| 23 | \( 1 + 0.828T + 23T^{2} \) |
| 29 | \( 1 - 2.82T + 29T^{2} \) |
| 31 | \( 1 - 2.82T + 31T^{2} \) |
| 37 | \( 1 + 9.65T + 37T^{2} \) |
| 41 | \( 1 + 1.75T + 41T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 - 12.4T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 - 8.48T + 59T^{2} \) |
| 61 | \( 1 + 3.07T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 + 6.48T + 71T^{2} \) |
| 73 | \( 1 + 16.2T + 73T^{2} \) |
| 79 | \( 1 + 2.34T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 + 14.2T + 89T^{2} \) |
| 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
−7.46498760222159273743634623649, −6.73819596163184691902359302712, −5.92261609293520518516079461205, −5.42270841971141695942205780070, −4.40438842005853880903787914455, −3.90354672070654207970195668352, −2.83912290502608412088540378646, −2.34776637748721021389676289526, −1.37582143557372265510835878909, 0,
1.37582143557372265510835878909, 2.34776637748721021389676289526, 2.83912290502608412088540378646, 3.90354672070654207970195668352, 4.40438842005853880903787914455, 5.42270841971141695942205780070, 5.92261609293520518516079461205, 6.73819596163184691902359302712, 7.46498760222159273743634623649
