| L(s) = 1 | − 5.32·2-s − 2.36·3-s + 12.3·4-s + 48.6·5-s + 12.5·6-s + 49·7-s + 19.5·8-s − 75.4·9-s − 259.·10-s − 29.1·12-s − 260.·14-s − 115.·15-s − 301.·16-s + 289·17-s + 401.·18-s + 599.·20-s − 115.·21-s − 46.3·24-s + 1.74e3·25-s + 369.·27-s + 603.·28-s + 612.·30-s − 1.19e3·31-s + 1.29e3·32-s − 1.53e3·34-s + 2.38e3·35-s − 929.·36-s + ⋯ |
| L(s) = 1 | − 1.33·2-s − 0.262·3-s + 0.770·4-s + 1.94·5-s + 0.349·6-s + 0.999·7-s + 0.305·8-s − 0.930·9-s − 2.59·10-s − 0.202·12-s − 1.33·14-s − 0.511·15-s − 1.17·16-s + 17-s + 1.23·18-s + 1.49·20-s − 0.262·21-s − 0.0804·24-s + 2.79·25-s + 0.507·27-s + 0.770·28-s + 0.680·30-s − 1.24·31-s + 1.26·32-s − 1.33·34-s + 1.94·35-s − 0.716·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.210600354\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.210600354\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 49T \) |
| 17 | \( 1 - 289T \) |
| good | 2 | \( 1 + 5.32T + 16T^{2} \) |
| 3 | \( 1 + 2.36T + 81T^{2} \) |
| 5 | \( 1 - 48.6T + 625T^{2} \) |
| 11 | \( 1 - 1.46e4T^{2} \) |
| 13 | \( 1 - 2.85e4T^{2} \) |
| 19 | \( 1 - 1.30e5T^{2} \) |
| 23 | \( 1 - 2.79e5T^{2} \) |
| 29 | \( 1 - 7.07e5T^{2} \) |
| 31 | \( 1 + 1.19e3T + 9.23e5T^{2} \) |
| 37 | \( 1 - 1.87e6T^{2} \) |
| 41 | \( 1 - 1.88e3T + 2.82e6T^{2} \) |
| 43 | \( 1 - 3.61e3T + 3.41e6T^{2} \) |
| 47 | \( 1 - 4.87e6T^{2} \) |
| 53 | \( 1 + 5.59e3T + 7.89e6T^{2} \) |
| 59 | \( 1 - 1.21e7T^{2} \) |
| 61 | \( 1 - 5.03e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 62.3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 2.54e7T^{2} \) |
| 73 | \( 1 + 447.T + 2.83e7T^{2} \) |
| 79 | \( 1 - 3.89e7T^{2} \) |
| 83 | \( 1 - 4.74e7T^{2} \) |
| 89 | \( 1 - 6.27e7T^{2} \) |
| 97 | \( 1 - 1.78e4T + 8.85e7T^{2} \) |
| 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
−12.77245277583608150885006658924, −11.21615532934697301172407805729, −10.51013388386865680761349838759, −9.498417398762011321949886783573, −8.790540826467837749698268582037, −7.59400528826169043530229664057, −6.06201440997441937226512756729, −5.13589916581228588106923377757, −2.30499547896858713097240513864, −1.13228314430001149775983219335,
1.13228314430001149775983219335, 2.30499547896858713097240513864, 5.13589916581228588106923377757, 6.06201440997441937226512756729, 7.59400528826169043530229664057, 8.790540826467837749698268582037, 9.498417398762011321949886783573, 10.51013388386865680761349838759, 11.21615532934697301172407805729, 12.77245277583608150885006658924
