| L(s) = 1 | − 2-s + 4-s − 6·5-s − 8-s + 9-s + 6·10-s − 8·13-s + 16-s + 6·17-s − 18-s − 6·20-s + 17·25-s + 8·26-s − 2·29-s − 32-s − 6·34-s + 36-s − 2·37-s + 6·40-s − 18·41-s − 6·45-s + 11·49-s − 17·50-s − 8·52-s − 12·53-s + 2·58-s − 20·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 2.68·5-s − 0.353·8-s + 1/3·9-s + 1.89·10-s − 2.21·13-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 1.34·20-s + 17/5·25-s + 1.56·26-s − 0.371·29-s − 0.176·32-s − 1.02·34-s + 1/6·36-s − 0.328·37-s + 0.948·40-s − 2.81·41-s − 0.894·45-s + 11/7·49-s − 2.40·50-s − 1.10·52-s − 1.64·53-s + 0.262·58-s − 2.56·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 242208 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 242208 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2230704226\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2230704226\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 29 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.742732893743880397661101853194, −8.443424828695416122862075486969, −7.83268146353528315500962133734, −7.53788920881883152221186875139, −7.43139202428287466808911876527, −6.97231192698558333316844758012, −6.29337110214496682069455772774, −5.40752116211565700486495950334, −4.85512283254279582558561488230, −4.50728162696553084669927927089, −3.70435541571971822099748167322, −3.37304116241702720928761290471, −2.74469725024707480151490295474, −1.63272357936029339660940403376, −0.32069404542737933413219678177,
0.32069404542737933413219678177, 1.63272357936029339660940403376, 2.74469725024707480151490295474, 3.37304116241702720928761290471, 3.70435541571971822099748167322, 4.50728162696553084669927927089, 4.85512283254279582558561488230, 5.40752116211565700486495950334, 6.29337110214496682069455772774, 6.97231192698558333316844758012, 7.43139202428287466808911876527, 7.53788920881883152221186875139, 7.83268146353528315500962133734, 8.443424828695416122862075486969, 8.742732893743880397661101853194
