| L(s) = 1 | + 4.30·2-s − 9·3-s − 13.4·4-s − 25·5-s − 38.7·6-s − 148.·7-s − 195.·8-s + 81·9-s − 107.·10-s + 121·11-s + 121.·12-s + 234.·13-s − 637.·14-s + 225·15-s − 409.·16-s + 1.03e3·17-s + 348.·18-s + 1.26e3·19-s + 337.·20-s + 1.33e3·21-s + 520.·22-s + 384.·23-s + 1.76e3·24-s + 625·25-s + 1.00e3·26-s − 729·27-s + 2.00e3·28-s + ⋯ |
| L(s) = 1 | + 0.760·2-s − 0.577·3-s − 0.421·4-s − 0.447·5-s − 0.438·6-s − 1.14·7-s − 1.08·8-s + 0.333·9-s − 0.340·10-s + 0.301·11-s + 0.243·12-s + 0.384·13-s − 0.869·14-s + 0.258·15-s − 0.400·16-s + 0.865·17-s + 0.253·18-s + 0.804·19-s + 0.188·20-s + 0.660·21-s + 0.229·22-s + 0.151·23-s + 0.624·24-s + 0.200·25-s + 0.292·26-s − 0.192·27-s + 0.482·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.311744798\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.311744798\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 9T \) |
| 5 | \( 1 + 25T \) |
| 11 | \( 1 - 121T \) |
| good | 2 | \( 1 - 4.30T + 32T^{2} \) |
| 7 | \( 1 + 148.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 234.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.03e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.26e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 384.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.48e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 997.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.16e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 2.25e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.58e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.20e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.85e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.02e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.00e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.09e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.69e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.66e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.85e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.22e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.51e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 9.93e4T + 8.58e9T^{2} \) |
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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
−12.21087539771608444998597708322, −11.19178267501443237409974314319, −9.852615066011022509109132191102, −9.073521128793540909720294743676, −7.54564376253530081571860023514, −6.27019943724201140548148196582, −5.43362080904759813289245081167, −4.07827055618879432479428933903, −3.18580952412537033892790519167, −0.67413957804236615491802169391,
0.67413957804236615491802169391, 3.18580952412537033892790519167, 4.07827055618879432479428933903, 5.43362080904759813289245081167, 6.27019943724201140548148196582, 7.54564376253530081571860023514, 9.073521128793540909720294743676, 9.852615066011022509109132191102, 11.19178267501443237409974314319, 12.21087539771608444998597708322
