| L(s) = 1 | − 21.4·2-s − 50.4·4-s + 44.8·5-s + 9.34e3·7-s + 1.20e4·8-s − 963.·10-s + 5.51e4·11-s − 7.99e4·13-s − 2.00e5·14-s − 2.33e5·16-s − 8.59e3·17-s + 6.43e5·19-s − 2.26e3·20-s − 1.18e6·22-s − 5.33e5·23-s − 1.95e6·25-s + 1.71e6·26-s − 4.71e5·28-s + 5.28e6·29-s − 3.34e6·31-s − 1.16e6·32-s + 1.84e5·34-s + 4.19e5·35-s − 2.03e7·37-s − 1.38e7·38-s + 5.41e5·40-s + 3.18e7·41-s + ⋯ |
| L(s) = 1 | − 0.949·2-s − 0.0985·4-s + 0.0320·5-s + 1.47·7-s + 1.04·8-s − 0.0304·10-s + 1.13·11-s − 0.776·13-s − 1.39·14-s − 0.891·16-s − 0.0249·17-s + 1.13·19-s − 0.00316·20-s − 1.07·22-s − 0.397·23-s − 0.998·25-s + 0.737·26-s − 0.145·28-s + 1.38·29-s − 0.650·31-s − 0.196·32-s + 0.0236·34-s + 0.0472·35-s − 1.78·37-s − 1.07·38-s + 0.0334·40-s + 1.75·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.403832308\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.403832308\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + 21.4T + 512T^{2} \) |
| 5 | \( 1 - 44.8T + 1.95e6T^{2} \) |
| 7 | \( 1 - 9.34e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 5.51e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 7.99e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 8.59e3T + 1.18e11T^{2} \) |
| 19 | \( 1 - 6.43e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 5.33e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 5.28e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 3.34e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 2.03e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 3.18e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 4.55e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.80e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 6.95e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 6.48e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 8.75e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.67e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.55e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.05e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 3.59e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 4.63e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 2.70e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.50e8T + 7.60e17T^{2} \) |
| 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
−12.17275033760370929016038864323, −11.28920519570137910448276458340, −10.09348397252683304608460871105, −9.098923705565670721262725139609, −8.105897765122539465735512518599, −7.17601016877490860349331990328, −5.26814276219777911678197770961, −4.12925762901531651788177046240, −1.90483784958485558506004807337, −0.856474554977397540443683623326,
0.856474554977397540443683623326, 1.90483784958485558506004807337, 4.12925762901531651788177046240, 5.26814276219777911678197770961, 7.17601016877490860349331990328, 8.105897765122539465735512518599, 9.098923705565670721262725139609, 10.09348397252683304608460871105, 11.28920519570137910448276458340, 12.17275033760370929016038864323
