| L(s) = 1 | + 2.02·2-s − 2.41·3-s + 2.10·4-s + 2.55·5-s − 4.90·6-s − 1.67·7-s + 0.220·8-s + 2.84·9-s + 5.17·10-s + 4.49·11-s − 5.09·12-s + 3.91·13-s − 3.40·14-s − 6.16·15-s − 3.77·16-s + 17-s + 5.76·18-s + 2.30·19-s + 5.37·20-s + 4.05·21-s + 9.11·22-s + 2.43·23-s − 0.532·24-s + 1.50·25-s + 7.92·26-s + 0.372·27-s − 3.53·28-s + ⋯ |
| L(s) = 1 | + 1.43·2-s − 1.39·3-s + 1.05·4-s + 1.14·5-s − 2.00·6-s − 0.634·7-s + 0.0778·8-s + 0.948·9-s + 1.63·10-s + 1.35·11-s − 1.47·12-s + 1.08·13-s − 0.909·14-s − 1.59·15-s − 0.942·16-s + 0.242·17-s + 1.35·18-s + 0.529·19-s + 1.20·20-s + 0.885·21-s + 1.94·22-s + 0.507·23-s − 0.108·24-s + 0.301·25-s + 1.55·26-s + 0.0716·27-s − 0.668·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.534843587\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.534843587\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 - T \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 - 2.02T + 2T^{2} \) |
| 3 | \( 1 + 2.41T + 3T^{2} \) |
| 5 | \( 1 - 2.55T + 5T^{2} \) |
| 7 | \( 1 + 1.67T + 7T^{2} \) |
| 11 | \( 1 - 4.49T + 11T^{2} \) |
| 13 | \( 1 - 3.91T + 13T^{2} \) |
| 19 | \( 1 - 2.30T + 19T^{2} \) |
| 23 | \( 1 - 2.43T + 23T^{2} \) |
| 29 | \( 1 - 5.50T + 29T^{2} \) |
| 31 | \( 1 - 1.31T + 31T^{2} \) |
| 37 | \( 1 - 7.98T + 37T^{2} \) |
| 41 | \( 1 + 4.96T + 41T^{2} \) |
| 43 | \( 1 + 2.30T + 43T^{2} \) |
| 53 | \( 1 - 5.59T + 53T^{2} \) |
| 59 | \( 1 + 3.88T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 + 12.1T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 + 0.754T + 73T^{2} \) |
| 79 | \( 1 + 2.66T + 79T^{2} \) |
| 83 | \( 1 + 6.18T + 83T^{2} \) |
| 89 | \( 1 - 1.29T + 89T^{2} \) |
| 97 | \( 1 + 7.40T + 97T^{2} \) |
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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
−10.47908963851026234282096066136, −9.655552564677094822863504412768, −8.783800403110693497368886199725, −6.88207455107838288295505474182, −6.26220556646623511268676482136, −5.93769432571390087130124282826, −5.07364449106812648403313003352, −4.09872791715528853665239012788, −3.00023164136869854344091960527, −1.27872030363248170497339462756,
1.27872030363248170497339462756, 3.00023164136869854344091960527, 4.09872791715528853665239012788, 5.07364449106812648403313003352, 5.93769432571390087130124282826, 6.26220556646623511268676482136, 6.88207455107838288295505474182, 8.783800403110693497368886199725, 9.655552564677094822863504412768, 10.47908963851026234282096066136
