| L(s) = 1 | − 2·3-s − 4-s − 2·5-s − 3·7-s + 9-s + 11-s + 2·12-s − 4·13-s + 4·15-s − 3·16-s + 2·17-s − 4·19-s + 2·20-s + 6·21-s + 3·23-s + 4·27-s + 3·28-s − 5·29-s − 2·33-s + 6·35-s − 36-s + 37-s + 8·39-s + 9·41-s + 43-s − 44-s − 2·45-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1/2·4-s − 0.894·5-s − 1.13·7-s + 1/3·9-s + 0.301·11-s + 0.577·12-s − 1.10·13-s + 1.03·15-s − 3/4·16-s + 0.485·17-s − 0.917·19-s + 0.447·20-s + 1.30·21-s + 0.625·23-s + 0.769·27-s + 0.566·28-s − 0.928·29-s − 0.348·33-s + 1.01·35-s − 1/6·36-s + 0.164·37-s + 1.28·39-s + 1.40·41-s + 0.152·43-s − 0.150·44-s − 0.298·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5049 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5049 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 3 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 - 3 T + 40 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 5 T + 28 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 52 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - T - 48 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - T - 18 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 3 T + 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 2 T - 38 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 5 T + 46 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 64 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 70 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $D_{4}$ | \( 1 - 2 T - 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $D_{4}$ | \( 1 + 5 T - 60 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
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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
−17.5791186300, −17.2423252843, −16.7211818031, −16.2488473424, −15.9934097183, −15.1781268198, −14.7835516268, −14.2309006824, −13.3832581669, −12.8609545312, −12.5413698115, −11.7810297743, −11.6203594754, −10.7885658945, −10.3549562131, −9.50615136877, −9.16540967581, −8.35677285456, −7.42754456813, −6.98264529743, −6.19801262985, −5.56057121158, −4.62424112982, −4.05268096818, −2.85941548054, 0,
2.85941548054, 4.05268096818, 4.62424112982, 5.56057121158, 6.19801262985, 6.98264529743, 7.42754456813, 8.35677285456, 9.16540967581, 9.50615136877, 10.3549562131, 10.7885658945, 11.6203594754, 11.7810297743, 12.5413698115, 12.8609545312, 13.3832581669, 14.2309006824, 14.7835516268, 15.1781268198, 15.9934097183, 16.2488473424, 16.7211818031, 17.2423252843, 17.5791186300
