| L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s + 6.79·5-s − 6·6-s + 7·7-s − 8·8-s + 9·9-s − 13.5·10-s + 14.2·11-s + 12·12-s + 66.3·13-s − 14·14-s + 20.3·15-s + 16·16-s + 60.4·17-s − 18·18-s − 19·19-s + 27.1·20-s + 21·21-s − 28.5·22-s − 66.7·23-s − 24·24-s − 78.8·25-s − 132.·26-s + 27·27-s + 28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.607·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.429·10-s + 0.391·11-s + 0.288·12-s + 1.41·13-s − 0.267·14-s + 0.350·15-s + 0.250·16-s + 0.861·17-s − 0.235·18-s − 0.229·19-s + 0.303·20-s + 0.218·21-s − 0.276·22-s − 0.605·23-s − 0.204·24-s − 0.630·25-s − 1.00·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.511327984\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.511327984\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 - 7T \) |
| 19 | \( 1 + 19T \) |
| good | 5 | \( 1 - 6.79T + 125T^{2} \) |
| 11 | \( 1 - 14.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 66.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 60.4T + 4.91e3T^{2} \) |
| 23 | \( 1 + 66.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 235.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 121.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 157.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 229.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 84.3T + 7.95e4T^{2} \) |
| 47 | \( 1 - 228.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 423.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 265.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 233.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 315.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 580.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 622.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 704.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 42.8T + 5.71e5T^{2} \) |
| 89 | \( 1 - 437.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.44e3T + 9.12e5T^{2} \) |
| 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
−9.784183955353046324315406928108, −8.995899066733106233292444889843, −8.291230271108866465037472381882, −7.57473272715648510026795038934, −6.39540729120940712088173512929, −5.74936226966965935971880954921, −4.27338412628207472918861811902, −3.18599975421829734842269098846, −1.94233688914449665965346673714, −1.04195574494949546001117906982,
1.04195574494949546001117906982, 1.94233688914449665965346673714, 3.18599975421829734842269098846, 4.27338412628207472918861811902, 5.74936226966965935971880954921, 6.39540729120940712088173512929, 7.57473272715648510026795038934, 8.291230271108866465037472381882, 8.995899066733106233292444889843, 9.784183955353046324315406928108
