| L(s) = 1 | + 2-s + 4-s + 3·7-s + 8-s + 9-s + 3·14-s + 16-s + 17-s + 18-s + 4·23-s − 2·25-s + 3·28-s − 5·31-s + 32-s + 34-s + 36-s + 4·41-s + 4·46-s − 2·47-s + 2·49-s − 2·50-s + 3·56-s − 5·62-s + 3·63-s + 64-s + 68-s + 10·71-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.13·7-s + 0.353·8-s + 1/3·9-s + 0.801·14-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.834·23-s − 2/5·25-s + 0.566·28-s − 0.898·31-s + 0.176·32-s + 0.171·34-s + 1/6·36-s + 0.624·41-s + 0.589·46-s − 0.291·47-s + 2/7·49-s − 0.282·50-s + 0.400·56-s − 0.635·62-s + 0.377·63-s + 1/8·64-s + 0.121·68-s + 1.18·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 106624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 106624 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.869763552\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.869763552\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 7 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 2 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 92 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
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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
−9.559407510236729900112316353403, −9.038166849861221898697776609294, −8.438020669954198689935731277871, −7.951587118363700113362999191452, −7.54666318908771424558431208255, −7.00957925929268447313796770479, −6.51408850554377511191127178576, −5.80888177295663579456518749268, −5.26294870630057927057680823628, −4.91549485206440347599637543290, −4.21212270803142914832608113986, −3.72985618660138208284445815215, −2.89229709510836272608842459911, −2.08024436888947667497213428535, −1.28086390105815995820121152303,
1.28086390105815995820121152303, 2.08024436888947667497213428535, 2.89229709510836272608842459911, 3.72985618660138208284445815215, 4.21212270803142914832608113986, 4.91549485206440347599637543290, 5.26294870630057927057680823628, 5.80888177295663579456518749268, 6.51408850554377511191127178576, 7.00957925929268447313796770479, 7.54666318908771424558431208255, 7.951587118363700113362999191452, 8.438020669954198689935731277871, 9.038166849861221898697776609294, 9.559407510236729900112316353403
