| L(s) = 1 | + 4·5-s − 9-s + 4·13-s − 4·17-s + 6·25-s + 4·29-s + 4·37-s − 12·41-s − 4·45-s + 6·49-s + 4·53-s − 12·61-s + 16·65-s − 12·73-s + 81-s − 16·85-s + 12·89-s + 12·97-s − 12·101-s − 12·109-s − 12·113-s − 4·117-s + 6·121-s + 4·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 1/3·9-s + 1.10·13-s − 0.970·17-s + 6/5·25-s + 0.742·29-s + 0.657·37-s − 1.87·41-s − 0.596·45-s + 6/7·49-s + 0.549·53-s − 1.53·61-s + 1.98·65-s − 1.40·73-s + 1/9·81-s − 1.73·85-s + 1.27·89-s + 1.21·97-s − 1.19·101-s − 1.14·109-s − 1.12·113-s − 0.369·117-s + 6/11·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.734381402\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.734381402\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 106 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−10.30833213514757498513882859120, −9.900179014472606279069626876746, −9.230886051609571412695073940227, −8.891445991295568381980794602378, −8.464816342473742258615278696129, −7.75750408276040742564631190481, −6.91950027314728133792020930159, −6.38561974216324161889860279739, −6.05037829844096332478399513944, −5.48430265702634826766610341478, −4.83828885667238315435619144403, −4.03658575092496091758835830863, −3.07868749500386681033869486735, −2.28708818518850225684617667408, −1.47130844084057074366336267008,
1.47130844084057074366336267008, 2.28708818518850225684617667408, 3.07868749500386681033869486735, 4.03658575092496091758835830863, 4.83828885667238315435619144403, 5.48430265702634826766610341478, 6.05037829844096332478399513944, 6.38561974216324161889860279739, 6.91950027314728133792020930159, 7.75750408276040742564631190481, 8.464816342473742258615278696129, 8.891445991295568381980794602378, 9.230886051609571412695073940227, 9.900179014472606279069626876746, 10.30833213514757498513882859120
