| L(s) = 1 | + 10.1·2-s + 9·3-s + 71.8·4-s − 25·5-s + 91.7·6-s + 134.·7-s + 405.·8-s + 81·9-s − 254.·10-s − 121·11-s + 646.·12-s + 328.·13-s + 1.37e3·14-s − 225·15-s + 1.83e3·16-s + 636.·17-s + 825.·18-s − 975.·19-s − 1.79e3·20-s + 1.21e3·21-s − 1.23e3·22-s − 1.14e3·23-s + 3.65e3·24-s + 625·25-s + 3.34e3·26-s + 729·27-s + 9.67e3·28-s + ⋯ |
| L(s) = 1 | + 1.80·2-s + 0.577·3-s + 2.24·4-s − 0.447·5-s + 1.04·6-s + 1.03·7-s + 2.24·8-s + 0.333·9-s − 0.805·10-s − 0.301·11-s + 1.29·12-s + 0.539·13-s + 1.87·14-s − 0.258·15-s + 1.79·16-s + 0.534·17-s + 0.600·18-s − 0.620·19-s − 1.00·20-s + 0.599·21-s − 0.543·22-s − 0.450·23-s + 1.29·24-s + 0.200·25-s + 0.971·26-s + 0.192·27-s + 2.33·28-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
| \(L(3)\) |
\(\approx\) |
\(7.293373444\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.293373444\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 10.1T + 32T^{2} \) |
| 7 | \( 1 - 134.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 328.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 636.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 975.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.14e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.07e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 8.56e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.88e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 9.04e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.49e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.53e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 5.13e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.36e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.57e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.12e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.28e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.21e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.26e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.01e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.67e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.80e5T + 8.58e9T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.11558659283266863433184707290, −11.34817891187257382187827291696, −10.35523657570035459749170917205, −8.470164994942807548999867276750, −7.58770339127314489512157139304, −6.33596746890887404497556443164, −5.06099052493172188027443866012, −4.17739749471187409121119654167, −3.06948516834859269655700726518, −1.72099424417437276491844059529,
1.72099424417437276491844059529, 3.06948516834859269655700726518, 4.17739749471187409121119654167, 5.06099052493172188027443866012, 6.33596746890887404497556443164, 7.58770339127314489512157139304, 8.470164994942807548999867276750, 10.35523657570035459749170917205, 11.34817891187257382187827291696, 12.11558659283266863433184707290
