| L(s) = 1 | + 2-s + 4-s + 5-s − 2·7-s + 8-s − 3·9-s + 10-s + 4·11-s − 2·13-s − 2·14-s + 16-s − 4·17-s − 3·18-s + 2·19-s + 20-s + 4·22-s − 5·25-s − 2·26-s − 2·28-s − 5·29-s + 32-s − 4·34-s − 2·35-s − 3·36-s + 9·37-s + 2·38-s + 40-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.755·7-s + 0.353·8-s − 9-s + 0.316·10-s + 1.20·11-s − 0.554·13-s − 0.534·14-s + 1/4·16-s − 0.970·17-s − 0.707·18-s + 0.458·19-s + 0.223·20-s + 0.852·22-s − 25-s − 0.392·26-s − 0.377·28-s − 0.928·29-s + 0.176·32-s − 0.685·34-s − 0.338·35-s − 1/2·36-s + 1.47·37-s + 0.324·38-s + 0.158·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.282334412\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.282334412\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( 1 - T \) |
| 433 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 5 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $C_4$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 2 T - 20 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 10 T + 66 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2 T + 46 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 79 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 67 | $D_{4}$ | \( 1 - 12 T + 122 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $D_{4}$ | \( 1 + 5 T + 46 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 151 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T - 28 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 15 T + 110 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 11 T + 94 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
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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
−17.0590755193, −16.4542771128, −16.1577554709, −15.5119079993, −14.8512386633, −14.4837249189, −14.1206298442, −13.4996256626, −12.9285018858, −12.6926512126, −11.6937550425, −11.5078087690, −11.0811910643, −10.0675107187, −9.54333498180, −9.22530287099, −8.37977778906, −7.61315676619, −6.84524274512, −6.19511265147, −5.85589843451, −4.90218119718, −4.02209370915, −3.18924885282, −2.16512744481,
2.16512744481, 3.18924885282, 4.02209370915, 4.90218119718, 5.85589843451, 6.19511265147, 6.84524274512, 7.61315676619, 8.37977778906, 9.22530287099, 9.54333498180, 10.0675107187, 11.0811910643, 11.5078087690, 11.6937550425, 12.6926512126, 12.9285018858, 13.4996256626, 14.1206298442, 14.4837249189, 14.8512386633, 15.5119079993, 16.1577554709, 16.4542771128, 17.0590755193
