| L(s) = 1 | − 2-s − 3-s − 2·4-s − 5·5-s + 6-s − 3·7-s + 3·8-s − 4·9-s + 5·10-s − 11-s + 2·12-s − 3·13-s + 3·14-s + 5·15-s + 16-s + 2·17-s + 4·18-s − 3·19-s + 10·20-s + 3·21-s + 22-s + 3·23-s − 3·24-s + 10·25-s + 3·26-s + 6·27-s + 6·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 4-s − 2.23·5-s + 0.408·6-s − 1.13·7-s + 1.06·8-s − 4/3·9-s + 1.58·10-s − 0.301·11-s + 0.577·12-s − 0.832·13-s + 0.801·14-s + 1.29·15-s + 1/4·16-s + 0.485·17-s + 0.942·18-s − 0.688·19-s + 2.23·20-s + 0.654·21-s + 0.213·22-s + 0.625·23-s − 0.612·24-s + 2·25-s + 0.588·26-s + 1.15·27-s + 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18329 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18329 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 18329 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 110 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + T + 5 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_4$ | \( 1 + p T + 3 p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 15 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 7 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 3 T + T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 3 T - 11 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 29 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + T + 16 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 3 T - 17 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 5 T + 52 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 5 T + 62 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 3 T - 49 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 17 T + 185 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 7 T + 60 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 71 | $D_{4}$ | \( 1 + 3 T - 41 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 9 T + 97 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 134 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 12 T + 154 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−16.5169800677, −15.9545277432, −15.3988122497, −15.0168113546, −14.6101608294, −13.9386101204, −13.3774704426, −12.7909386331, −12.3441587954, −11.9936895215, −11.4013930668, −10.9637059183, −10.5443220452, −9.65306912230, −9.28303833710, −8.81301827934, −8.16366492202, −7.73654314440, −7.37329621404, −6.44525534596, −5.71518244527, −4.97329336831, −4.31296192354, −3.59670565912, −2.97392193068, 0, 0,
2.97392193068, 3.59670565912, 4.31296192354, 4.97329336831, 5.71518244527, 6.44525534596, 7.37329621404, 7.73654314440, 8.16366492202, 8.81301827934, 9.28303833710, 9.65306912230, 10.5443220452, 10.9637059183, 11.4013930668, 11.9936895215, 12.3441587954, 12.7909386331, 13.3774704426, 13.9386101204, 14.6101608294, 15.0168113546, 15.3988122497, 15.9545277432, 16.5169800677
