| L(s) = 1 | + 8.07·2-s − 9·3-s + 33.2·4-s − 72.7·6-s + 39.3·7-s + 10.2·8-s + 81·9-s + 121·11-s − 299.·12-s + 220.·13-s + 318.·14-s − 981.·16-s + 200.·17-s + 654.·18-s + 350.·19-s − 354.·21-s + 977.·22-s − 1.38e3·23-s − 91.9·24-s + 1.78e3·26-s − 729·27-s + 1.30e3·28-s + 5.50e3·29-s − 2.45e3·31-s − 8.25e3·32-s − 1.08e3·33-s + 1.61e3·34-s + ⋯ |
| L(s) = 1 | + 1.42·2-s − 0.577·3-s + 1.03·4-s − 0.824·6-s + 0.303·7-s + 0.0564·8-s + 0.333·9-s + 0.301·11-s − 0.600·12-s + 0.362·13-s + 0.433·14-s − 0.958·16-s + 0.168·17-s + 0.476·18-s + 0.222·19-s − 0.175·21-s + 0.430·22-s − 0.545·23-s − 0.0325·24-s + 0.517·26-s − 0.192·27-s + 0.315·28-s + 1.21·29-s − 0.458·31-s − 1.42·32-s − 0.174·33-s + 0.240·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.186492961\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.186492961\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 - 121T \) |
| good | 2 | \( 1 - 8.07T + 32T^{2} \) |
| 7 | \( 1 - 39.3T + 1.68e4T^{2} \) |
| 13 | \( 1 - 220.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 200.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 350.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.38e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.50e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.45e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.06e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 527.T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.20e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 563.T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.72e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.15e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.99e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.84e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.37e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.97e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.56e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.99e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.77e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 7.61e3T + 8.58e9T^{2} \) |
| 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
−9.593396194088515146350330895503, −8.580990740455532707459821914283, −7.45922811635295521071980971882, −6.46829463597262933770636335296, −5.85234625320562815312527535114, −4.97774192926619066629652996187, −4.23026222599976917878515372944, −3.33680405051253071304348989548, −2.13069831831111957379794882914, −0.77309700318192769304041189439,
0.77309700318192769304041189439, 2.13069831831111957379794882914, 3.33680405051253071304348989548, 4.23026222599976917878515372944, 4.97774192926619066629652996187, 5.85234625320562815312527535114, 6.46829463597262933770636335296, 7.45922811635295521071980971882, 8.580990740455532707459821914283, 9.593396194088515146350330895503
