| L(s) = 1 | + 0.286·2-s − 1.75·3-s − 1.91·4-s − 2.61·5-s − 0.504·6-s − 1.12·8-s + 0.0962·9-s − 0.749·10-s + 5.95·11-s + 3.37·12-s + 4.60·15-s + 3.51·16-s + 0.218·17-s + 0.0275·18-s + 6.05·19-s + 5.01·20-s + 1.70·22-s + 1.77·23-s + 1.97·24-s + 1.83·25-s + 5.10·27-s + 9.23·29-s + 1.31·30-s + 6.25·31-s + 3.25·32-s − 10.4·33-s + 0.0627·34-s + ⋯ |
| L(s) = 1 | + 0.202·2-s − 1.01·3-s − 0.958·4-s − 1.16·5-s − 0.205·6-s − 0.396·8-s + 0.0320·9-s − 0.236·10-s + 1.79·11-s + 0.974·12-s + 1.18·15-s + 0.878·16-s + 0.0530·17-s + 0.00650·18-s + 1.38·19-s + 1.12·20-s + 0.364·22-s + 0.369·23-s + 0.403·24-s + 0.367·25-s + 0.983·27-s + 1.71·29-s + 0.240·30-s + 1.12·31-s + 0.574·32-s − 1.82·33-s + 0.0107·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9701183595\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9701183595\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.286T + 2T^{2} \) |
| 3 | \( 1 + 1.75T + 3T^{2} \) |
| 5 | \( 1 + 2.61T + 5T^{2} \) |
| 11 | \( 1 - 5.95T + 11T^{2} \) |
| 17 | \( 1 - 0.218T + 17T^{2} \) |
| 19 | \( 1 - 6.05T + 19T^{2} \) |
| 23 | \( 1 - 1.77T + 23T^{2} \) |
| 29 | \( 1 - 9.23T + 29T^{2} \) |
| 31 | \( 1 - 6.25T + 31T^{2} \) |
| 37 | \( 1 + 5.85T + 37T^{2} \) |
| 41 | \( 1 + 2.74T + 41T^{2} \) |
| 43 | \( 1 - 0.366T + 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 - 7.58T + 53T^{2} \) |
| 59 | \( 1 + 2.44T + 59T^{2} \) |
| 61 | \( 1 + 9.37T + 61T^{2} \) |
| 67 | \( 1 - 12.3T + 67T^{2} \) |
| 71 | \( 1 - 1.05T + 71T^{2} \) |
| 73 | \( 1 + 6.01T + 73T^{2} \) |
| 79 | \( 1 - 12.4T + 79T^{2} \) |
| 83 | \( 1 + 7.46T + 83T^{2} \) |
| 89 | \( 1 - 0.418T + 89T^{2} \) |
| 97 | \( 1 - 13.3T + 97T^{2} \) |
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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
−7.88726675132400919237446637250, −6.88929585848643884995339702809, −6.46869718491687678377833051633, −5.63645385641571269916098153604, −4.85252355509094325664335333583, −4.46271846815874922339919537520, −3.59762743247096873338121758896, −3.11224044977355524311155831544, −1.22792732656473752062678490685, −0.60005227751602343240879082516,
0.60005227751602343240879082516, 1.22792732656473752062678490685, 3.11224044977355524311155831544, 3.59762743247096873338121758896, 4.46271846815874922339919537520, 4.85252355509094325664335333583, 5.63645385641571269916098153604, 6.46869718491687678377833051633, 6.88929585848643884995339702809, 7.88726675132400919237446637250
