| L(s) = 1 | − 4.98·2-s + 6.26·3-s + 16.8·4-s + 15.7·5-s − 31.1·6-s + 0.789·7-s − 43.8·8-s + 12.2·9-s − 78.4·10-s + 45.3·11-s + 105.·12-s + 46.7·13-s − 3.93·14-s + 98.6·15-s + 84.1·16-s − 60.8·18-s + 100.·19-s + 264.·20-s + 4.94·21-s − 225.·22-s − 84.1·23-s − 274.·24-s + 122.·25-s − 232.·26-s − 92.5·27-s + 13.2·28-s − 101.·29-s + ⋯ |
| L(s) = 1 | − 1.76·2-s + 1.20·3-s + 2.10·4-s + 1.40·5-s − 2.12·6-s + 0.0426·7-s − 1.93·8-s + 0.452·9-s − 2.48·10-s + 1.24·11-s + 2.53·12-s + 0.997·13-s − 0.0751·14-s + 1.69·15-s + 1.31·16-s − 0.796·18-s + 1.21·19-s + 2.95·20-s + 0.0514·21-s − 2.18·22-s − 0.762·23-s − 2.33·24-s + 0.983·25-s − 1.75·26-s − 0.659·27-s + 0.0896·28-s − 0.651·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.750298730\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.750298730\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 + 4.98T + 8T^{2} \) |
| 3 | \( 1 - 6.26T + 27T^{2} \) |
| 5 | \( 1 - 15.7T + 125T^{2} \) |
| 7 | \( 1 - 0.789T + 343T^{2} \) |
| 11 | \( 1 - 45.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 46.7T + 2.19e3T^{2} \) |
| 19 | \( 1 - 100.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 84.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 101.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 7.36T + 2.97e4T^{2} \) |
| 37 | \( 1 + 251.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 260.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 401.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 304.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 398.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 577.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 126.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 150.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 434.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 493.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 72.2T + 4.93e5T^{2} \) |
| 83 | \( 1 - 711.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.35e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.40e3T + 9.12e5T^{2} \) |
| 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
−10.94908962419047493715278843560, −9.880909613709864675577745366284, −9.349390177081718831091476571831, −8.808248997215767261528088177712, −7.911177341063065465398213070275, −6.78681765735450841983551937532, −5.83231206244244182018787334091, −3.46429478832676728086777939234, −2.10845157839168448932791045071, −1.31083311754023641881283385196,
1.31083311754023641881283385196, 2.10845157839168448932791045071, 3.46429478832676728086777939234, 5.83231206244244182018787334091, 6.78681765735450841983551937532, 7.911177341063065465398213070275, 8.808248997215767261528088177712, 9.349390177081718831091476571831, 9.880909613709864675577745366284, 10.94908962419047493715278843560
