| L(s) = 1 | − 1.24·2-s − 2.67·3-s − 0.449·4-s + 2.61·5-s + 3.33·6-s + 1.20·7-s + 3.05·8-s + 4.17·9-s − 3.26·10-s + 4.39·11-s + 1.20·12-s + 1.21·13-s − 1.49·14-s − 7.01·15-s − 2.89·16-s − 3.65·17-s − 5.19·18-s + 6.42·19-s − 1.17·20-s − 3.22·21-s − 5.47·22-s − 3.94·23-s − 8.17·24-s + 1.85·25-s − 1.51·26-s − 3.14·27-s − 0.541·28-s + ⋯ |
| L(s) = 1 | − 0.880·2-s − 1.54·3-s − 0.224·4-s + 1.17·5-s + 1.36·6-s + 0.455·7-s + 1.07·8-s + 1.39·9-s − 1.03·10-s + 1.32·11-s + 0.347·12-s + 0.337·13-s − 0.400·14-s − 1.81·15-s − 0.724·16-s − 0.887·17-s − 1.22·18-s + 1.47·19-s − 0.263·20-s − 0.704·21-s − 1.16·22-s − 0.822·23-s − 1.66·24-s + 0.371·25-s − 0.297·26-s − 0.605·27-s − 0.102·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7488263852\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7488263852\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 \) |
| good | 2 | \( 1 + 1.24T + 2T^{2} \) |
| 3 | \( 1 + 2.67T + 3T^{2} \) |
| 5 | \( 1 - 2.61T + 5T^{2} \) |
| 7 | \( 1 - 1.20T + 7T^{2} \) |
| 11 | \( 1 - 4.39T + 11T^{2} \) |
| 13 | \( 1 - 1.21T + 13T^{2} \) |
| 17 | \( 1 + 3.65T + 17T^{2} \) |
| 19 | \( 1 - 6.42T + 19T^{2} \) |
| 23 | \( 1 + 3.94T + 23T^{2} \) |
| 29 | \( 1 - 1.34T + 29T^{2} \) |
| 37 | \( 1 + 6.43T + 37T^{2} \) |
| 41 | \( 1 - 2.08T + 41T^{2} \) |
| 43 | \( 1 - 9.41T + 43T^{2} \) |
| 47 | \( 1 - 0.754T + 47T^{2} \) |
| 53 | \( 1 - 4.71T + 53T^{2} \) |
| 59 | \( 1 - 3.99T + 59T^{2} \) |
| 61 | \( 1 - 2.21T + 61T^{2} \) |
| 67 | \( 1 - 12.0T + 67T^{2} \) |
| 71 | \( 1 + 7.52T + 71T^{2} \) |
| 73 | \( 1 + 9.92T + 73T^{2} \) |
| 79 | \( 1 - 9.46T + 79T^{2} \) |
| 83 | \( 1 + 5.60T + 83T^{2} \) |
| 89 | \( 1 - 13.9T + 89T^{2} \) |
| 97 | \( 1 + 4.50T + 97T^{2} \) |
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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
−9.981246691049411667677166778504, −9.379148537552071086282148752813, −8.629782968909435870905125194005, −7.38475284856362018486399424357, −6.53291440127388491136374789248, −5.77362662090925271482338345678, −5.00147968080813824806780557983, −4.04401794593612549614824122338, −1.81624023867207721185602335240, −0.911767111508828267462815132741,
0.911767111508828267462815132741, 1.81624023867207721185602335240, 4.04401794593612549614824122338, 5.00147968080813824806780557983, 5.77362662090925271482338345678, 6.53291440127388491136374789248, 7.38475284856362018486399424357, 8.629782968909435870905125194005, 9.379148537552071086282148752813, 9.981246691049411667677166778504
