| L(s) = 1 | − 2-s − 4·3-s + 4-s + 4·6-s + 2·7-s − 8-s + 6·9-s − 4·12-s − 8·13-s − 2·14-s + 16-s − 6·18-s − 8·21-s + 4·24-s − 10·25-s + 8·26-s + 4·27-s + 2·28-s − 12·29-s − 32-s + 6·36-s + 32·39-s + 8·42-s − 4·48-s + 3·49-s + 10·50-s − 8·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 2.30·3-s + 1/2·4-s + 1.63·6-s + 0.755·7-s − 0.353·8-s + 2·9-s − 1.15·12-s − 2.21·13-s − 0.534·14-s + 1/4·16-s − 1.41·18-s − 1.74·21-s + 0.816·24-s − 2·25-s + 1.56·26-s + 0.769·27-s + 0.377·28-s − 2.22·29-s − 0.176·32-s + 36-s + 5.12·39-s + 1.23·42-s − 0.577·48-s + 3/7·49-s + 1.41·50-s − 1.10·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 758912 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 758912 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_2$ | \( 1 + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−7.57571100088867902110310233811, −7.40807001536020266346628728051, −7.03079318663473438367614947672, −6.34180261438630109455978749892, −5.99101346792442609117915659634, −5.57928681742950427486583645839, −5.14686312215284501903612495216, −4.94441063949591210730742590235, −4.26529121158947013398640035047, −3.64206819084684017249230081826, −2.58665972235577718718744868923, −2.12283190546293616976391637892, −1.28535617009511908054229014087, 0, 0,
1.28535617009511908054229014087, 2.12283190546293616976391637892, 2.58665972235577718718744868923, 3.64206819084684017249230081826, 4.26529121158947013398640035047, 4.94441063949591210730742590235, 5.14686312215284501903612495216, 5.57928681742950427486583645839, 5.99101346792442609117915659634, 6.34180261438630109455978749892, 7.03079318663473438367614947672, 7.40807001536020266346628728051, 7.57571100088867902110310233811
