| L(s) = 1 | + 2.62·3-s + 0.309·5-s + 4.59·7-s + 3.87·9-s + 5.71·11-s + 1.28·13-s + 0.810·15-s + 2.73·19-s + 12.0·21-s + 1.53·23-s − 4.90·25-s + 2.28·27-s − 5.05·29-s − 4.39·31-s + 14.9·33-s + 1.42·35-s + 10.4·37-s + 3.37·39-s + 1.94·41-s − 10.1·43-s + 1.19·45-s − 10.7·47-s + 14.1·49-s + 6.19·53-s + 1.76·55-s + 7.16·57-s + 11.9·59-s + ⋯ |
| L(s) = 1 | + 1.51·3-s + 0.138·5-s + 1.73·7-s + 1.29·9-s + 1.72·11-s + 0.356·13-s + 0.209·15-s + 0.627·19-s + 2.62·21-s + 0.319·23-s − 0.980·25-s + 0.438·27-s − 0.938·29-s − 0.789·31-s + 2.60·33-s + 0.240·35-s + 1.71·37-s + 0.540·39-s + 0.303·41-s − 1.55·43-s + 0.178·45-s − 1.56·47-s + 2.01·49-s + 0.850·53-s + 0.238·55-s + 0.949·57-s + 1.55·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.797987656\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.797987656\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - 2.62T + 3T^{2} \) |
| 5 | \( 1 - 0.309T + 5T^{2} \) |
| 7 | \( 1 - 4.59T + 7T^{2} \) |
| 11 | \( 1 - 5.71T + 11T^{2} \) |
| 13 | \( 1 - 1.28T + 13T^{2} \) |
| 19 | \( 1 - 2.73T + 19T^{2} \) |
| 23 | \( 1 - 1.53T + 23T^{2} \) |
| 29 | \( 1 + 5.05T + 29T^{2} \) |
| 31 | \( 1 + 4.39T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 - 1.94T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 - 6.19T + 53T^{2} \) |
| 59 | \( 1 - 11.9T + 59T^{2} \) |
| 61 | \( 1 + 8.42T + 61T^{2} \) |
| 67 | \( 1 - 13.4T + 67T^{2} \) |
| 71 | \( 1 - 0.548T + 71T^{2} \) |
| 73 | \( 1 + 7.46T + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 + 5.50T + 83T^{2} \) |
| 89 | \( 1 - 11.8T + 89T^{2} \) |
| 97 | \( 1 + 2.70T + 97T^{2} \) |
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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
−7.73511111313617462549481253107, −7.39055804468747788836216786449, −6.47836750331573245781168062977, −5.59404683721818872042491945022, −4.78286298871800557881400991929, −3.93239639109568880530909139041, −3.64370362211842239977500659270, −2.50420182754364167926909714365, −1.69818247200975521275306190334, −1.28416602114394556790743163756,
1.28416602114394556790743163756, 1.69818247200975521275306190334, 2.50420182754364167926909714365, 3.64370362211842239977500659270, 3.93239639109568880530909139041, 4.78286298871800557881400991929, 5.59404683721818872042491945022, 6.47836750331573245781168062977, 7.39055804468747788836216786449, 7.73511111313617462549481253107
