| L(s) = 1 | + 0.390·2-s − 0.150·3-s − 1.84·4-s + 1.62·5-s − 0.0586·6-s − 1.50·8-s − 2.97·9-s + 0.633·10-s + 0.277·12-s + 6.47·13-s − 0.243·15-s + 3.10·16-s + 4.38·17-s − 1.16·18-s + 4.11·19-s − 2.99·20-s − 5.37·23-s + 0.225·24-s − 2.37·25-s + 2.53·26-s + 0.896·27-s − 5.92·29-s − 0.0950·30-s + 1.38·31-s + 4.22·32-s + 1.71·34-s + 5.50·36-s + ⋯ |
| L(s) = 1 | + 0.276·2-s − 0.0866·3-s − 0.923·4-s + 0.724·5-s − 0.0239·6-s − 0.531·8-s − 0.992·9-s + 0.200·10-s + 0.0800·12-s + 1.79·13-s − 0.0627·15-s + 0.776·16-s + 1.06·17-s − 0.274·18-s + 0.944·19-s − 0.669·20-s − 1.12·23-s + 0.0460·24-s − 0.474·25-s + 0.496·26-s + 0.172·27-s − 1.10·29-s − 0.0173·30-s + 0.249·31-s + 0.746·32-s + 0.293·34-s + 0.916·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5929 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5929 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.925846150\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.925846150\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.390T + 2T^{2} \) |
| 3 | \( 1 + 0.150T + 3T^{2} \) |
| 5 | \( 1 - 1.62T + 5T^{2} \) |
| 13 | \( 1 - 6.47T + 13T^{2} \) |
| 17 | \( 1 - 4.38T + 17T^{2} \) |
| 19 | \( 1 - 4.11T + 19T^{2} \) |
| 23 | \( 1 + 5.37T + 23T^{2} \) |
| 29 | \( 1 + 5.92T + 29T^{2} \) |
| 31 | \( 1 - 1.38T + 31T^{2} \) |
| 37 | \( 1 - 1.62T + 37T^{2} \) |
| 41 | \( 1 - 0.566T + 41T^{2} \) |
| 43 | \( 1 - 6.93T + 43T^{2} \) |
| 47 | \( 1 - 7.62T + 47T^{2} \) |
| 53 | \( 1 + 8.36T + 53T^{2} \) |
| 59 | \( 1 + 9.96T + 59T^{2} \) |
| 61 | \( 1 + 2.87T + 61T^{2} \) |
| 67 | \( 1 + 7.57T + 67T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 + 1.35T + 73T^{2} \) |
| 79 | \( 1 + 15.9T + 79T^{2} \) |
| 83 | \( 1 + 3.49T + 83T^{2} \) |
| 89 | \( 1 - 14.0T + 89T^{2} \) |
| 97 | \( 1 - 11.2T + 97T^{2} \) |
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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
−8.083234374393857777402644089846, −7.63157639398835746992259193243, −6.19616779464014625415285142837, −5.82514637003100679143510639643, −5.52144921459609239471967509780, −4.42698827376173820300867139808, −3.59918746436583280431399320767, −3.07190451343681046942057396905, −1.76256241988789772854234106415, −0.73243733462245679591128569650,
0.73243733462245679591128569650, 1.76256241988789772854234106415, 3.07190451343681046942057396905, 3.59918746436583280431399320767, 4.42698827376173820300867139808, 5.52144921459609239471967509780, 5.82514637003100679143510639643, 6.19616779464014625415285142837, 7.63157639398835746992259193243, 8.083234374393857777402644089846
