L(s) = 1 | − 10.6·2-s + 22.7·3-s + 81.4·4-s + 25·5-s − 242.·6-s − 527.·8-s + 273.·9-s − 266.·10-s + 521.·11-s + 1.85e3·12-s − 530.·13-s + 567.·15-s + 3.00e3·16-s − 1.99e3·17-s − 2.90e3·18-s − 2.49e3·19-s + 2.03e3·20-s − 5.55e3·22-s − 3.55e3·23-s − 1.19e4·24-s + 625·25-s + 5.65e3·26-s + 683.·27-s − 4.28e3·29-s − 6.05e3·30-s − 5.03e3·31-s − 1.51e4·32-s + ⋯ |
L(s) = 1 | − 1.88·2-s + 1.45·3-s + 2.54·4-s + 0.447·5-s − 2.74·6-s − 2.91·8-s + 1.12·9-s − 0.842·10-s + 1.29·11-s + 3.71·12-s − 0.870·13-s + 0.651·15-s + 2.93·16-s − 1.67·17-s − 2.11·18-s − 1.58·19-s + 1.13·20-s − 2.44·22-s − 1.40·23-s − 4.24·24-s + 0.200·25-s + 1.63·26-s + 0.180·27-s − 0.946·29-s − 1.22·30-s − 0.941·31-s − 2.62·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - 25T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 10.6T + 32T^{2} \) |
| 3 | \( 1 - 22.7T + 243T^{2} \) |
| 11 | \( 1 - 521.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 530.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.99e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.49e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.55e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.28e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.03e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 2.62e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.82e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 5.08e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 4.98e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.90e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 8.93e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.35e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.37e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.69e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 9.79e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.85e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.55e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.12e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.56e5T + 8.58e9T^{2} \) |
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
−10.29896399429853867458435316756, −9.356730669961253205410156705700, −8.962783851080409203412274712710, −8.196897543727946088169630769010, −7.13891250229380713210440020338, −6.33761255936027313181792858028, −3.90935019377357245408814128373, −2.25462267332343100229731910683, −1.89592597830198172918582047616, 0,
1.89592597830198172918582047616, 2.25462267332343100229731910683, 3.90935019377357245408814128373, 6.33761255936027313181792858028, 7.13891250229380713210440020338, 8.196897543727946088169630769010, 8.962783851080409203412274712710, 9.356730669961253205410156705700, 10.29896399429853867458435316756