| L(s) = 1 | − 1.92·2-s − 9·3-s − 28.2·4-s + 17.3·6-s + 59.1·7-s + 116.·8-s + 81·9-s − 121·11-s + 254.·12-s − 599.·13-s − 113.·14-s + 682.·16-s + 2.27e3·17-s − 155.·18-s − 2.99e3·19-s − 532.·21-s + 232.·22-s + 1.57e3·23-s − 1.04e3·24-s + 1.15e3·26-s − 729·27-s − 1.67e3·28-s + 9.02e3·29-s + 1.83e3·31-s − 5.02e3·32-s + 1.08e3·33-s − 4.37e3·34-s + ⋯ |
| L(s) = 1 | − 0.340·2-s − 0.577·3-s − 0.884·4-s + 0.196·6-s + 0.455·7-s + 0.641·8-s + 0.333·9-s − 0.301·11-s + 0.510·12-s − 0.984·13-s − 0.155·14-s + 0.666·16-s + 1.90·17-s − 0.113·18-s − 1.90·19-s − 0.263·21-s + 0.102·22-s + 0.619·23-s − 0.370·24-s + 0.334·26-s − 0.192·27-s − 0.403·28-s + 1.99·29-s + 0.343·31-s − 0.867·32-s + 0.174·33-s − 0.648·34-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
| \(L(3)\) |
\(\approx\) |
\(0.8997009739\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8997009739\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 1.92T + 32T^{2} \) |
| 7 | \( 1 - 59.1T + 1.68e4T^{2} \) |
| 13 | \( 1 + 599.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.27e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.99e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.57e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 9.02e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.83e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 3.88e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 2.14e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.61e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 8.37e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.12e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.50e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.28e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 8.32e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.34e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.94e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.04e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 8.21e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 3.84e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.39e5T + 8.58e9T^{2} \) |
| 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
−9.632249531729967868029725102197, −8.505931282417219050210173533472, −7.971735546643385654251368216850, −6.98865691925276867572811247764, −5.84836692280294202190180363814, −4.92107154516101017267205881967, −4.40434583464631671920141281098, −3.01005132576675512610223140724, −1.52989506334743613380374036920, −0.49786675485691329818516983671,
0.49786675485691329818516983671, 1.52989506334743613380374036920, 3.01005132576675512610223140724, 4.40434583464631671920141281098, 4.92107154516101017267205881967, 5.84836692280294202190180363814, 6.98865691925276867572811247764, 7.971735546643385654251368216850, 8.505931282417219050210173533472, 9.632249531729967868029725102197
