| L(s) = 1 | − 3.22·2-s + 2.34·3-s + 2.37·4-s + 11.1·5-s − 7.55·6-s + 7·7-s + 18.1·8-s − 21.4·9-s − 36.0·10-s + 5.57·12-s + 35.6·13-s − 22.5·14-s + 26.2·15-s − 77.3·16-s − 19.6·17-s + 69.2·18-s + 58.5·19-s + 26.5·20-s + 16.4·21-s + 37.8·23-s + 42.5·24-s − 0.0216·25-s − 114.·26-s − 113.·27-s + 16.6·28-s + 244.·29-s − 84.5·30-s + ⋯ |
| L(s) = 1 | − 1.13·2-s + 0.451·3-s + 0.296·4-s + 0.999·5-s − 0.514·6-s + 0.377·7-s + 0.800·8-s − 0.795·9-s − 1.13·10-s + 0.134·12-s + 0.760·13-s − 0.430·14-s + 0.451·15-s − 1.20·16-s − 0.280·17-s + 0.906·18-s + 0.707·19-s + 0.296·20-s + 0.170·21-s + 0.343·23-s + 0.361·24-s − 0.000173·25-s − 0.866·26-s − 0.811·27-s + 0.112·28-s + 1.56·29-s − 0.514·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.560071358\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.560071358\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 7T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 3.22T + 8T^{2} \) |
| 3 | \( 1 - 2.34T + 27T^{2} \) |
| 5 | \( 1 - 11.1T + 125T^{2} \) |
| 13 | \( 1 - 35.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 19.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 58.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 37.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 244.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 8.24T + 2.97e4T^{2} \) |
| 37 | \( 1 - 117.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 143.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 483.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 368.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 571.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 57.4T + 2.05e5T^{2} \) |
| 61 | \( 1 + 806.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 532.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 119.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 975.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 286.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 491.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.24e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 132.T + 9.12e5T^{2} \) |
| 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
−9.466770796027933044715407186028, −9.097884456270909590036135471560, −8.294934633020554352886615250834, −7.62670423851240182197425399989, −6.43091742267189277912689226348, −5.55016725043876104726279301541, −4.43470378035558527020846013189, −2.96432318448902660999052504522, −1.89003960840299121679304185447, −0.835181770520087971482624683275,
0.835181770520087971482624683275, 1.89003960840299121679304185447, 2.96432318448902660999052504522, 4.43470378035558527020846013189, 5.55016725043876104726279301541, 6.43091742267189277912689226348, 7.62670423851240182197425399989, 8.294934633020554352886615250834, 9.097884456270909590036135471560, 9.466770796027933044715407186028
