L(s) = 1 | − 3.33·2-s + 9·3-s − 20.8·4-s − 30.0·6-s + 103.·7-s + 176.·8-s + 81·9-s + 121·11-s − 187.·12-s − 357.·13-s − 345.·14-s + 78.7·16-s + 2.06e3·17-s − 270.·18-s + 1.32e3·19-s + 931.·21-s − 403.·22-s − 2.49e3·23-s + 1.58e3·24-s + 1.19e3·26-s + 729·27-s − 2.15e3·28-s + 1.65e3·29-s + 4.69e3·31-s − 5.90e3·32-s + 1.08e3·33-s − 6.90e3·34-s + ⋯ |
L(s) = 1 | − 0.589·2-s + 0.577·3-s − 0.651·4-s − 0.340·6-s + 0.798·7-s + 0.974·8-s + 0.333·9-s + 0.301·11-s − 0.376·12-s − 0.587·13-s − 0.470·14-s + 0.0769·16-s + 1.73·17-s − 0.196·18-s + 0.838·19-s + 0.460·21-s − 0.177·22-s − 0.984·23-s + 0.562·24-s + 0.346·26-s + 0.192·27-s − 0.520·28-s + 0.366·29-s + 0.876·31-s − 1.01·32-s + 0.174·33-s − 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.168680016\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.168680016\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 9T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - 121T \) |
good | 2 | \( 1 + 3.33T + 32T^{2} \) |
| 7 | \( 1 - 103.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 357.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.06e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.32e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.49e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.65e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.69e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.21e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.35e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.07e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.92e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.15e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.54e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.64e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.54e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.49e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.35e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.91e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.05e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.11e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 5.82e4T + 8.58e9T^{2} \) |
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
−9.660919068603323869972036899762, −8.557779528567587504814695676225, −7.83773449774469824351068272731, −7.46608644315081240445528778698, −5.90843792353154713285006805235, −4.90281949418899243826719479876, −4.10673702799699721959935964041, −2.95562605983844381788354165623, −1.59460467851108882611484848166, −0.78062203901852461632972930591,
0.78062203901852461632972930591, 1.59460467851108882611484848166, 2.95562605983844381788354165623, 4.10673702799699721959935964041, 4.90281949418899243826719479876, 5.90843792353154713285006805235, 7.46608644315081240445528778698, 7.83773449774469824351068272731, 8.557779528567587504814695676225, 9.660919068603323869972036899762