| L(s) = 1 | − 3·3-s − 4-s − 2·5-s − 4·7-s + 2·9-s + 2·11-s + 3·12-s + 13-s + 6·15-s − 3·16-s + 2·17-s + 3·19-s + 2·20-s + 12·21-s − 23-s + 6·27-s + 4·28-s − 12·29-s − 6·33-s + 8·35-s − 2·36-s − 5·37-s − 3·39-s − 41-s + 3·43-s − 2·44-s − 4·45-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1/2·4-s − 0.894·5-s − 1.51·7-s + 2/3·9-s + 0.603·11-s + 0.866·12-s + 0.277·13-s + 1.54·15-s − 3/4·16-s + 0.485·17-s + 0.688·19-s + 0.447·20-s + 2.61·21-s − 0.208·23-s + 1.15·27-s + 0.755·28-s − 2.22·29-s − 1.04·33-s + 1.35·35-s − 1/3·36-s − 0.821·37-s − 0.480·39-s − 0.156·41-s + 0.457·43-s − 0.301·44-s − 0.596·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3037 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3037 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3037 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 10 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $D_{4}$ | \( 1 - T + 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 3 T - 3 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + T + 42 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 + 5 T + 43 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + T - 20 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 3 T - 10 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 9 T + 80 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 5 T + p T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 13 T + 108 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 14 T + 136 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $D_{4}$ | \( 1 + 5 T - 74 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 5 T - 43 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−18.4895335551, −17.8292580645, −17.3934806167, −16.8875859793, −16.3847379716, −16.0217712549, −15.7244814727, −14.7622081631, −14.2808731505, −13.4027521275, −12.9787441725, −12.3863827289, −11.7843402318, −11.3728603100, −11.0723638905, −10.0531589564, −9.58922054210, −8.92060057067, −8.07688803317, −7.02633161621, −6.63105027803, −5.73402192868, −5.32537623724, −4.10996113837, −3.37640430301, 0,
3.37640430301, 4.10996113837, 5.32537623724, 5.73402192868, 6.63105027803, 7.02633161621, 8.07688803317, 8.92060057067, 9.58922054210, 10.0531589564, 11.0723638905, 11.3728603100, 11.7843402318, 12.3863827289, 12.9787441725, 13.4027521275, 14.2808731505, 14.7622081631, 15.7244814727, 16.0217712549, 16.3847379716, 16.8875859793, 17.3934806167, 17.8292580645, 18.4895335551
