| L(s) = 1 | − 2·2-s − 4·3-s + 4-s − 2·5-s + 8·6-s − 3·7-s + 7·9-s + 4·10-s − 6·11-s − 4·12-s − 3·13-s + 6·14-s + 8·15-s + 16-s + 17-s − 14·18-s − 7·19-s − 2·20-s + 12·21-s + 12·22-s + 25-s + 6·26-s − 4·27-s − 3·28-s + 2·29-s − 16·30-s − 8·31-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 2.30·3-s + 1/2·4-s − 0.894·5-s + 3.26·6-s − 1.13·7-s + 7/3·9-s + 1.26·10-s − 1.80·11-s − 1.15·12-s − 0.832·13-s + 1.60·14-s + 2.06·15-s + 1/4·16-s + 0.242·17-s − 3.29·18-s − 1.60·19-s − 0.447·20-s + 2.61·21-s + 2.55·22-s + 1/5·25-s + 1.17·26-s − 0.769·27-s − 0.566·28-s + 0.371·29-s − 2.92·30-s − 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7697 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7697 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 4 T + p T^{2} ) \) |
| 179 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 16 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $D_{4}$ | \( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 9 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 3 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - T + 15 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 7 T + 40 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2 T + 21 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 64 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 61 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - T - 67 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 14 T + 121 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 16 T + 144 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 12 T + 76 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 3 T + 109 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 14 T + 150 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2 T + 85 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 5 T - 69 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 18 T + 244 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 5 T + 76 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
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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
−17.3674846385, −16.9217741159, −16.6134884446, −16.3303524406, −15.7959847774, −15.1150435316, −14.9410851572, −13.6556925084, −12.9896812789, −12.6212386857, −12.1638272105, −11.6935128779, −11.0108019165, −10.6210451909, −10.1894403437, −9.79337215941, −8.88572388007, −8.37537534172, −7.59452626591, −7.09770425464, −6.26487855717, −5.76770721973, −5.07809553259, −4.30253868145, −2.86869545303, 0, 0,
2.86869545303, 4.30253868145, 5.07809553259, 5.76770721973, 6.26487855717, 7.09770425464, 7.59452626591, 8.37537534172, 8.88572388007, 9.79337215941, 10.1894403437, 10.6210451909, 11.0108019165, 11.6935128779, 12.1638272105, 12.6212386857, 12.9896812789, 13.6556925084, 14.9410851572, 15.1150435316, 15.7959847774, 16.3303524406, 16.6134884446, 16.9217741159, 17.3674846385
