| L(s) = 1 | − 2-s − 4-s − 2·5-s + 7-s + 3·8-s − 2·9-s + 2·10-s + 3·11-s + 2·13-s − 14-s − 16-s − 5·17-s + 2·18-s − 2·19-s + 2·20-s − 3·22-s − 4·23-s − 25-s − 2·26-s − 28-s + 3·29-s − 5·31-s − 5·32-s + 5·34-s − 2·35-s + 2·36-s − 6·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.894·5-s + 0.377·7-s + 1.06·8-s − 2/3·9-s + 0.632·10-s + 0.904·11-s + 0.554·13-s − 0.267·14-s − 1/4·16-s − 1.21·17-s + 0.471·18-s − 0.458·19-s + 0.447·20-s − 0.639·22-s − 0.834·23-s − 1/5·25-s − 0.392·26-s − 0.188·28-s + 0.557·29-s − 0.898·31-s − 0.883·32-s + 0.857·34-s − 0.338·35-s + 1/3·36-s − 0.986·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_2$ | \( 1 + T + p T^{2} \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $D_{4}$ | \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 3 T + 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 5 T + 16 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_4$ | \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 3 T - 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 5 T + 60 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $D_{4}$ | \( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + T - 48 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 3 T + 16 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - T + 84 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 11 T + 52 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 10 T + 78 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $D_{4}$ | \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2 T + 66 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 6 T - 62 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
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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
−14.7173496758, −14.1321721195, −13.8848663933, −13.4223994296, −12.8411449025, −12.3581711920, −11.8525894563, −11.4440519777, −11.0271626827, −10.5724447861, −10.2001391792, −9.31688312464, −8.92369977165, −8.82930535996, −8.16352157138, −7.71702164659, −7.27836350156, −6.55664170289, −5.98503167130, −5.35748332529, −4.41377393531, −4.18370295147, −3.62381527497, −2.46867912938, −1.42909995173, 0,
1.42909995173, 2.46867912938, 3.62381527497, 4.18370295147, 4.41377393531, 5.35748332529, 5.98503167130, 6.55664170289, 7.27836350156, 7.71702164659, 8.16352157138, 8.82930535996, 8.92369977165, 9.31688312464, 10.2001391792, 10.5724447861, 11.0271626827, 11.4440519777, 11.8525894563, 12.3581711920, 12.8411449025, 13.4223994296, 13.8848663933, 14.1321721195, 14.7173496758
