| L(s) = 1 | − 3·2-s − 2·3-s + 4·4-s − 4·5-s + 6·6-s − 2·7-s − 3·8-s + 4·9-s + 12·10-s − 6·11-s − 8·12-s − 5·13-s + 6·14-s + 8·15-s + 3·16-s − 4·17-s − 12·18-s − 4·19-s − 16·20-s + 4·21-s + 18·22-s − 7·23-s + 6·24-s + 6·25-s + 15·26-s − 5·27-s − 8·28-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.15·3-s + 2·4-s − 1.78·5-s + 2.44·6-s − 0.755·7-s − 1.06·8-s + 4/3·9-s + 3.79·10-s − 1.80·11-s − 2.30·12-s − 1.38·13-s + 1.60·14-s + 2.06·15-s + 3/4·16-s − 0.970·17-s − 2.82·18-s − 0.917·19-s − 3.57·20-s + 0.872·21-s + 3.83·22-s − 1.45·23-s + 1.22·24-s + 6/5·25-s + 2.94·26-s − 0.962·27-s − 1.51·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11751 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11751 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T + p T^{2} ) \) |
| 3917 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 18 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $D_{4}$ | \( 1 + 2 T + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 5 T + 25 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 25 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 7 T + 49 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 24 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 + T + 39 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 69 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 2 T + 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 10 T + 119 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 52 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 3 T + 93 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - T + 32 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - T + 66 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 2 T + 20 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 8 T - 13 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 3 T - 26 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 63 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−17.1906010246, −16.3387241116, −16.0539543686, −15.7750045455, −15.3268005316, −14.9901419044, −13.8342678411, −13.2158716639, −12.4548469272, −12.1446901810, −11.9956933159, −10.9422482215, −10.6830956363, −10.2294661114, −9.89511134529, −9.13218409236, −8.43376090238, −8.06215779716, −7.39776865617, −7.20921954800, −6.36342699799, −5.40659674953, −4.60668815418, −3.84179981192, −2.42220944466, 0, 0,
2.42220944466, 3.84179981192, 4.60668815418, 5.40659674953, 6.36342699799, 7.20921954800, 7.39776865617, 8.06215779716, 8.43376090238, 9.13218409236, 9.89511134529, 10.2294661114, 10.6830956363, 10.9422482215, 11.9956933159, 12.1446901810, 12.4548469272, 13.2158716639, 13.8342678411, 14.9901419044, 15.3268005316, 15.7750045455, 16.0539543686, 16.3387241116, 17.1906010246
