| L(s) = 1 | + 2-s − 2·3-s − 5-s − 2·6-s − 4·7-s + 8-s + 2·9-s − 10-s − 3·11-s − 2·13-s − 4·14-s + 2·15-s − 16-s + 6·17-s + 2·18-s − 2·19-s + 8·21-s − 3·22-s − 3·23-s − 2·24-s − 7·25-s − 2·26-s − 6·27-s + 3·29-s + 2·30-s − 2·31-s − 6·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s − 0.447·5-s − 0.816·6-s − 1.51·7-s + 0.353·8-s + 2/3·9-s − 0.316·10-s − 0.904·11-s − 0.554·13-s − 1.06·14-s + 0.516·15-s − 1/4·16-s + 1.45·17-s + 0.471·18-s − 0.458·19-s + 1.74·21-s − 0.639·22-s − 0.625·23-s − 0.408·24-s − 7/5·25-s − 0.392·26-s − 1.15·27-s + 0.557·29-s + 0.365·30-s − 0.359·31-s − 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14663 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14663 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + T + p T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 6 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $D_{4}$ | \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 19 | $D_{4}$ | \( 1 + 2 T + p T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 3 T + 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 - 5 T - 4 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 7 T + 110 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 16 T + 137 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + p T^{2} ) \) |
| 67 | $D_{4}$ | \( 1 + 11 T + 98 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $D_{4}$ | \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 202 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2 T - 113 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−16.1961827177, −15.9423202412, −15.6711500507, −14.9530730383, −14.3547354785, −13.7748599026, −13.3816309792, −12.8417795720, −12.5046999032, −11.8829871495, −11.7822514552, −10.8091286978, −10.4397566608, −9.94915444975, −9.52448376439, −8.65772758797, −7.80462584672, −7.32582191900, −6.78217166982, −5.96068035614, −5.51353085379, −5.06486523424, −3.94346368277, −3.67206512696, −2.36518571778, 0,
2.36518571778, 3.67206512696, 3.94346368277, 5.06486523424, 5.51353085379, 5.96068035614, 6.78217166982, 7.32582191900, 7.80462584672, 8.65772758797, 9.52448376439, 9.94915444975, 10.4397566608, 10.8091286978, 11.7822514552, 11.8829871495, 12.5046999032, 12.8417795720, 13.3816309792, 13.7748599026, 14.3547354785, 14.9530730383, 15.6711500507, 15.9423202412, 16.1961827177
