| L(s) = 1 | + 2.49·2-s − 1.76·4-s − 14.5·5-s − 24.3·8-s − 36.4·10-s + 0.0794·11-s + 86.9·13-s − 46.7·16-s − 93.0·17-s − 147.·19-s + 25.7·20-s + 0.198·22-s − 154.·23-s + 87.7·25-s + 217.·26-s − 205.·29-s + 271.·31-s + 78.3·32-s − 232.·34-s + 48.3·37-s − 369.·38-s + 355.·40-s − 52.3·41-s + 48.0·43-s − 0.140·44-s − 385.·46-s + 266.·47-s + ⋯ |
| L(s) = 1 | + 0.882·2-s − 0.220·4-s − 1.30·5-s − 1.07·8-s − 1.15·10-s + 0.00217·11-s + 1.85·13-s − 0.730·16-s − 1.32·17-s − 1.78·19-s + 0.288·20-s + 0.00192·22-s − 1.39·23-s + 0.701·25-s + 1.63·26-s − 1.31·29-s + 1.57·31-s + 0.433·32-s − 1.17·34-s + 0.214·37-s − 1.57·38-s + 1.40·40-s − 0.199·41-s + 0.170·43-s − 0.000480·44-s − 1.23·46-s + 0.825·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.265710995\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.265710995\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 2.49T + 8T^{2} \) |
| 5 | \( 1 + 14.5T + 125T^{2} \) |
| 11 | \( 1 - 0.0794T + 1.33e3T^{2} \) |
| 13 | \( 1 - 86.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 93.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 147.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 154.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 205.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 271.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 48.3T + 5.06e4T^{2} \) |
| 41 | \( 1 + 52.3T + 6.89e4T^{2} \) |
| 43 | \( 1 - 48.0T + 7.95e4T^{2} \) |
| 47 | \( 1 - 266.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 11.8T + 1.48e5T^{2} \) |
| 59 | \( 1 - 542.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 84.4T + 2.26e5T^{2} \) |
| 67 | \( 1 + 51.0T + 3.00e5T^{2} \) |
| 71 | \( 1 + 495.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 71.1T + 3.89e5T^{2} \) |
| 79 | \( 1 - 378.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 486.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.16e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.31e3T + 9.12e5T^{2} \) |
| 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
−8.805056219961435275507089807709, −8.623107020817190272249883541820, −7.72777139858724960695027213290, −6.41066747098203271141415344243, −6.07450786208484904407011297856, −4.68933798565594129492180339396, −3.99626344071598312466980614954, −3.67180858392766680597733582261, −2.23286718895847929345462738290, −0.47755916501800836198591919688,
0.47755916501800836198591919688, 2.23286718895847929345462738290, 3.67180858392766680597733582261, 3.99626344071598312466980614954, 4.68933798565594129492180339396, 6.07450786208484904407011297856, 6.41066747098203271141415344243, 7.72777139858724960695027213290, 8.623107020817190272249883541820, 8.805056219961435275507089807709
