| L(s) = 1 | − 2·2-s − 3·3-s + 4-s − 6·5-s + 6·6-s − 2·7-s + 2·8-s + 2·9-s + 12·10-s − 3·11-s − 3·12-s − 2·13-s + 4·14-s + 18·15-s − 3·16-s − 4·18-s − 5·19-s − 6·20-s + 6·21-s + 6·22-s − 23-s − 6·24-s + 18·25-s + 4·26-s + 6·27-s − 2·28-s − 5·29-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.73·3-s + 1/2·4-s − 2.68·5-s + 2.44·6-s − 0.755·7-s + 0.707·8-s + 2/3·9-s + 3.79·10-s − 0.904·11-s − 0.866·12-s − 0.554·13-s + 1.06·14-s + 4.64·15-s − 3/4·16-s − 0.942·18-s − 1.14·19-s − 1.34·20-s + 1.30·21-s + 1.27·22-s − 0.208·23-s − 1.22·24-s + 18/5·25-s + 0.784·26-s + 1.15·27-s − 0.377·28-s − 0.928·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6718 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6718 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_2$ | \( ( 1 + T )( 1 + T + p T^{2} ) \) |
| 3359 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 62 T + p T^{2} ) \) |
| good | 3 | $D_{4}$ | \( 1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 + 3 T + 16 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 5 T + 17 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + T + 20 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 5 T + 9 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 3 T + 13 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 8 T + 61 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 5 T + 45 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - T + 18 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 10 T + 85 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 3 T + 79 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - T + 59 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 9 T + 97 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 2 T + 52 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 15 T + 133 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 6 T + 63 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 7 T + 10 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + T - 69 T^{2} + p T^{3} + p^{2} T^{4} \) |
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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
−17.4206459208, −17.0986489738, −16.7151091865, −16.3108262918, −15.8738651421, −15.4061876675, −15.0207617296, −14.2258282824, −13.1568210697, −12.6938018761, −12.1791423988, −11.6594730938, −11.3854359465, −10.7776091507, −10.4068923888, −9.79433392685, −8.76878230440, −8.41305000415, −7.79309768958, −7.34349743875, −6.66886944676, −5.77200915563, −4.87221917305, −4.24933040685, −3.23736166881, 0, 0,
3.23736166881, 4.24933040685, 4.87221917305, 5.77200915563, 6.66886944676, 7.34349743875, 7.79309768958, 8.41305000415, 8.76878230440, 9.79433392685, 10.4068923888, 10.7776091507, 11.3854359465, 11.6594730938, 12.1791423988, 12.6938018761, 13.1568210697, 14.2258282824, 15.0207617296, 15.4061876675, 15.8738651421, 16.3108262918, 16.7151091865, 17.0986489738, 17.4206459208
