| L(s) = 1 | + 0.794·2-s − 12.4·3-s − 15.3·4-s − 25.8·5-s − 9.85·6-s − 49·7-s − 24.9·8-s + 72.8·9-s − 20.5·10-s + 190.·12-s − 38.9·14-s + 321.·15-s + 226.·16-s − 289·17-s + 57.8·18-s + 398.·20-s + 607.·21-s + 309.·24-s + 45.7·25-s + 101.·27-s + 753.·28-s + 255.·30-s − 1.06e3·31-s + 578.·32-s − 229.·34-s + 1.26e3·35-s − 1.11e3·36-s + ⋯ |
| L(s) = 1 | + 0.198·2-s − 1.37·3-s − 0.960·4-s − 1.03·5-s − 0.273·6-s − 0.999·7-s − 0.389·8-s + 0.898·9-s − 0.205·10-s + 1.32·12-s − 0.198·14-s + 1.42·15-s + 0.883·16-s − 17-s + 0.178·18-s + 0.995·20-s + 1.37·21-s + 0.536·24-s + 0.0732·25-s + 0.139·27-s + 0.960·28-s + 0.283·30-s − 1.10·31-s + 0.564·32-s − 0.198·34-s + 1.03·35-s − 0.863·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.2248027566\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2248027566\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + 49T \) |
| 17 | \( 1 + 289T \) |
| good | 2 | \( 1 - 0.794T + 16T^{2} \) |
| 3 | \( 1 + 12.4T + 81T^{2} \) |
| 5 | \( 1 + 25.8T + 625T^{2} \) |
| 11 | \( 1 - 1.46e4T^{2} \) |
| 13 | \( 1 - 2.85e4T^{2} \) |
| 19 | \( 1 - 1.30e5T^{2} \) |
| 23 | \( 1 - 2.79e5T^{2} \) |
| 29 | \( 1 - 7.07e5T^{2} \) |
| 31 | \( 1 + 1.06e3T + 9.23e5T^{2} \) |
| 37 | \( 1 - 1.87e6T^{2} \) |
| 41 | \( 1 + 116.T + 2.82e6T^{2} \) |
| 43 | \( 1 + 2.45e3T + 3.41e6T^{2} \) |
| 47 | \( 1 - 4.87e6T^{2} \) |
| 53 | \( 1 + 1.30e3T + 7.89e6T^{2} \) |
| 59 | \( 1 - 1.21e7T^{2} \) |
| 61 | \( 1 - 7.29e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 8.51e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 2.54e7T^{2} \) |
| 73 | \( 1 - 5.89e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 3.89e7T^{2} \) |
| 83 | \( 1 - 4.74e7T^{2} \) |
| 89 | \( 1 - 6.27e7T^{2} \) |
| 97 | \( 1 - 39.2T + 8.85e7T^{2} \) |
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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
−12.64774023207891151904048542857, −11.85550643226240237261817943370, −10.89661399005745333913422777583, −9.739855457086075751692008132954, −8.536186171025417220146012218885, −7.03467125075965685902194334241, −5.89622746509511433980313745250, −4.71818078484032232610058284887, −3.61078825164698487611901078283, −0.36086325087826941615964875111,
0.36086325087826941615964875111, 3.61078825164698487611901078283, 4.71818078484032232610058284887, 5.89622746509511433980313745250, 7.03467125075965685902194334241, 8.536186171025417220146012218885, 9.739855457086075751692008132954, 10.89661399005745333913422777583, 11.85550643226240237261817943370, 12.64774023207891151904048542857
