| L(s) = 1 | + 2.43·3-s − 1.82·5-s + 2.92·9-s + 3.78·11-s − 13-s − 4.44·15-s − 7.71·17-s − 1.44·19-s − 2.02·23-s − 1.66·25-s − 0.193·27-s − 0.611·29-s − 7.15·31-s + 9.19·33-s + 7.66·37-s − 2.43·39-s − 8.67·41-s − 2.13·43-s − 5.33·45-s + 4.69·47-s − 18.7·51-s + 5.24·53-s − 6.90·55-s − 3.50·57-s − 3.11·59-s + 10.2·61-s + 1.82·65-s + ⋯ |
| L(s) = 1 | + 1.40·3-s − 0.816·5-s + 0.973·9-s + 1.13·11-s − 0.277·13-s − 1.14·15-s − 1.87·17-s − 0.330·19-s − 0.421·23-s − 0.332·25-s − 0.0372·27-s − 0.113·29-s − 1.28·31-s + 1.60·33-s + 1.25·37-s − 0.389·39-s − 1.35·41-s − 0.325·43-s − 0.795·45-s + 0.684·47-s − 2.62·51-s + 0.720·53-s − 0.931·55-s − 0.464·57-s − 0.406·59-s + 1.31·61-s + 0.226·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 2.43T + 3T^{2} \) |
| 5 | \( 1 + 1.82T + 5T^{2} \) |
| 11 | \( 1 - 3.78T + 11T^{2} \) |
| 17 | \( 1 + 7.71T + 17T^{2} \) |
| 19 | \( 1 + 1.44T + 19T^{2} \) |
| 23 | \( 1 + 2.02T + 23T^{2} \) |
| 29 | \( 1 + 0.611T + 29T^{2} \) |
| 31 | \( 1 + 7.15T + 31T^{2} \) |
| 37 | \( 1 - 7.66T + 37T^{2} \) |
| 41 | \( 1 + 8.67T + 41T^{2} \) |
| 43 | \( 1 + 2.13T + 43T^{2} \) |
| 47 | \( 1 - 4.69T + 47T^{2} \) |
| 53 | \( 1 - 5.24T + 53T^{2} \) |
| 59 | \( 1 + 3.11T + 59T^{2} \) |
| 61 | \( 1 - 10.2T + 61T^{2} \) |
| 67 | \( 1 + 10.0T + 67T^{2} \) |
| 71 | \( 1 + 3.66T + 71T^{2} \) |
| 73 | \( 1 + 14.9T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 - 1.02T + 83T^{2} \) |
| 89 | \( 1 - 6.82T + 89T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
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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
−7.944914536426736263169900480939, −7.27769711072157340635311958051, −6.71219181988982744694515316125, −5.77480537487970355806809065468, −4.42388193336453212038589362937, −4.10231724367333174950511097515, −3.36717297686132964228398428076, −2.41173961402802904388961503936, −1.68036904676329825622704562893, 0,
1.68036904676329825622704562893, 2.41173961402802904388961503936, 3.36717297686132964228398428076, 4.10231724367333174950511097515, 4.42388193336453212038589362937, 5.77480537487970355806809065468, 6.71219181988982744694515316125, 7.27769711072157340635311958051, 7.944914536426736263169900480939
