| L(s) = 1 | + 0.131·2-s − 1.32·3-s − 1.98·4-s − 1.78·5-s − 0.173·6-s − 0.0149·7-s − 0.523·8-s − 1.25·9-s − 0.234·10-s − 2.07·11-s + 2.61·12-s − 1.62·13-s − 0.00196·14-s + 2.35·15-s + 3.89·16-s − 6.03·17-s − 0.164·18-s − 7.62·19-s + 3.53·20-s + 0.0197·21-s − 0.273·22-s − 5.30·23-s + 0.691·24-s − 1.82·25-s − 0.213·26-s + 5.62·27-s + 0.0297·28-s + ⋯ |
| L(s) = 1 | + 0.0929·2-s − 0.762·3-s − 0.991·4-s − 0.796·5-s − 0.0708·6-s − 0.00566·7-s − 0.185·8-s − 0.418·9-s − 0.0739·10-s − 0.627·11-s + 0.755·12-s − 0.450·13-s − 0.000526·14-s + 0.607·15-s + 0.974·16-s − 1.46·17-s − 0.0388·18-s − 1.74·19-s + 0.789·20-s + 0.00431·21-s − 0.0582·22-s − 1.10·23-s + 0.141·24-s − 0.365·25-s − 0.0418·26-s + 1.08·27-s + 0.00561·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1632557862\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1632557862\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 - 0.131T + 2T^{2} \) |
| 3 | \( 1 + 1.32T + 3T^{2} \) |
| 5 | \( 1 + 1.78T + 5T^{2} \) |
| 7 | \( 1 + 0.0149T + 7T^{2} \) |
| 11 | \( 1 + 2.07T + 11T^{2} \) |
| 13 | \( 1 + 1.62T + 13T^{2} \) |
| 17 | \( 1 + 6.03T + 17T^{2} \) |
| 19 | \( 1 + 7.62T + 19T^{2} \) |
| 23 | \( 1 + 5.30T + 23T^{2} \) |
| 29 | \( 1 - 7.17T + 29T^{2} \) |
| 31 | \( 1 - 2.38T + 31T^{2} \) |
| 37 | \( 1 + 5.04T + 37T^{2} \) |
| 41 | \( 1 - 3.55T + 41T^{2} \) |
| 47 | \( 1 + 10.0T + 47T^{2} \) |
| 53 | \( 1 + 6.85T + 53T^{2} \) |
| 59 | \( 1 + 1.66T + 59T^{2} \) |
| 61 | \( 1 - 6.05T + 61T^{2} \) |
| 67 | \( 1 - 1.43T + 67T^{2} \) |
| 71 | \( 1 - 7.04T + 71T^{2} \) |
| 73 | \( 1 + 9.76T + 73T^{2} \) |
| 79 | \( 1 - 12.0T + 79T^{2} \) |
| 83 | \( 1 - 9.06T + 83T^{2} \) |
| 89 | \( 1 - 15.9T + 89T^{2} \) |
| 97 | \( 1 - 13.5T + 97T^{2} \) |
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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
−9.137967647045760692644497560174, −8.292430262417704592741452164233, −8.017210921440512342096850379493, −6.61927806694972006431226596508, −6.09607549299636917886895779499, −4.87036055546788414043033495847, −4.59374599905130098672779821636, −3.57656749471169254855272041078, −2.29201878057915004281088786149, −0.26436723409594508435449339330,
0.26436723409594508435449339330, 2.29201878057915004281088786149, 3.57656749471169254855272041078, 4.59374599905130098672779821636, 4.87036055546788414043033495847, 6.09607549299636917886895779499, 6.61927806694972006431226596508, 8.017210921440512342096850379493, 8.292430262417704592741452164233, 9.137967647045760692644497560174
