| L(s) = 1 | − 3·2-s − 2·3-s + 4·4-s − 3·5-s + 6·6-s − 7·7-s − 3·8-s + 9-s + 9·10-s − 6·11-s − 8·12-s + 2·13-s + 21·14-s + 6·15-s + 3·16-s − 2·17-s − 3·18-s − 19-s − 12·20-s + 14·21-s + 18·22-s − 23-s + 6·24-s + 4·25-s − 6·26-s − 2·27-s − 28·28-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.15·3-s + 2·4-s − 1.34·5-s + 2.44·6-s − 2.64·7-s − 1.06·8-s + 1/3·9-s + 2.84·10-s − 1.80·11-s − 2.30·12-s + 0.554·13-s + 5.61·14-s + 1.54·15-s + 3/4·16-s − 0.485·17-s − 0.707·18-s − 0.229·19-s − 2.68·20-s + 3.05·21-s + 3.83·22-s − 0.208·23-s + 1.22·24-s + 4/5·25-s − 1.17·26-s − 0.384·27-s − 5.29·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7549 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7549 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 7549 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 148 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $D_{4}$ | \( 1 + 3 T + p T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + p T + 25 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 + T + 5 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 5 T + 57 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 7 T + 48 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 20 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 60 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 10 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 5 T + 67 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 10 T + 62 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 88 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 2 T + 21 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 23 T + 264 T^{2} - 23 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 9 T + 145 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 13 T + 167 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 3 T + 62 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
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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
−17.6233326206, −16.8133130710, −16.5649884160, −16.1074698367, −15.8783710153, −15.5109459209, −14.9266677000, −13.6384825781, −13.0815140719, −12.7666062662, −12.1823939437, −11.6233289252, −10.9026823072, −10.4738312018, −10.2020774276, −9.44023370596, −9.11497988036, −8.29184524629, −7.86950415435, −7.20654689735, −6.49212146811, −6.02827429272, −5.07765493599, −3.71332903296, −2.97641785817, 0, 0,
2.97641785817, 3.71332903296, 5.07765493599, 6.02827429272, 6.49212146811, 7.20654689735, 7.86950415435, 8.29184524629, 9.11497988036, 9.44023370596, 10.2020774276, 10.4738312018, 10.9026823072, 11.6233289252, 12.1823939437, 12.7666062662, 13.0815140719, 13.6384825781, 14.9266677000, 15.5109459209, 15.8783710153, 16.1074698367, 16.5649884160, 16.8133130710, 17.6233326206
