| L(s) = 1 | − 2-s + 4-s − 6·5-s − 8-s + 6·10-s − 2·13-s + 16-s − 6·17-s − 6·20-s + 17·25-s + 2·26-s + 18·29-s − 32-s + 6·34-s − 2·37-s + 6·40-s + 12·41-s + 2·49-s − 17·50-s − 2·52-s − 12·53-s − 18·58-s − 2·61-s + 64-s + 12·65-s − 6·68-s + 22·73-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 2.68·5-s − 0.353·8-s + 1.89·10-s − 0.554·13-s + 1/4·16-s − 1.45·17-s − 1.34·20-s + 17/5·25-s + 0.392·26-s + 3.34·29-s − 0.176·32-s + 1.02·34-s − 0.328·37-s + 0.948·40-s + 1.87·41-s + 2/7·49-s − 2.40·50-s − 0.277·52-s − 1.64·53-s − 2.36·58-s − 0.256·61-s + 1/8·64-s + 1.48·65-s − 0.727·68-s + 2.57·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 209952 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209952 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( 1 + T \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.542414927665885183690210552564, −8.278948375699901432058512087738, −8.047221009677602422962699873621, −7.44571478799237129140639120122, −7.16277862972928710155421440227, −6.50974172394367970294419909175, −6.29261335708839240986736404399, −5.10609983558079824652464594116, −4.60529471992634973001671480527, −4.25638665319375785757127060430, −3.68824329939728705086964001391, −2.94651413139897301734292877776, −2.41615112845105918353887046447, −0.935395355311356120178486862505, 0,
0.935395355311356120178486862505, 2.41615112845105918353887046447, 2.94651413139897301734292877776, 3.68824329939728705086964001391, 4.25638665319375785757127060430, 4.60529471992634973001671480527, 5.10609983558079824652464594116, 6.29261335708839240986736404399, 6.50974172394367970294419909175, 7.16277862972928710155421440227, 7.44571478799237129140639120122, 8.047221009677602422962699873621, 8.278948375699901432058512087738, 8.542414927665885183690210552564
