| L(s) = 1 | + 2·2-s + 2·4-s − 8·7-s + 4·8-s − 16·14-s + 8·16-s − 20·17-s + 4·23-s − 6·25-s − 16·28-s + 8·31-s + 8·32-s − 40·34-s − 10·41-s + 8·46-s − 12·47-s + 30·49-s − 12·50-s − 32·56-s + 16·62-s + 8·64-s − 40·68-s + 24·71-s + 36·73-s + 28·79-s − 20·82-s + 56·89-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 3.02·7-s + 1.41·8-s − 4.27·14-s + 2·16-s − 4.85·17-s + 0.834·23-s − 6/5·25-s − 3.02·28-s + 1.43·31-s + 1.41·32-s − 6.85·34-s − 1.56·41-s + 1.17·46-s − 1.75·47-s + 30/7·49-s − 1.69·50-s − 4.27·56-s + 2.03·62-s + 64-s − 4.85·68-s + 2.84·71-s + 4.21·73-s + 3.15·79-s − 2.20·82-s + 5.93·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.451904940\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.451904940\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \) |
| 11 | $C_2^3$ | \( 1 + 13 T^{2} + 48 T^{4} + 13 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )( 1 + 23 T^{2} + p^{2} T^{4} ) \) |
| 17 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 2 T - 19 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 5 T - 16 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 - 35 T^{2} - 624 T^{4} - 35 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 59 | $C_2^3$ | \( 1 + 117 T^{2} + 10208 T^{4} + 117 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} )( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} ) \) |
| 67 | $C_2^3$ | \( 1 + 125 T^{2} + 11136 T^{4} + 125 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 14 T + 117 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 150 T^{2} + 15611 T^{4} + 150 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 + T - 96 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.032662994130590356893921312244, −9.021931076140402793235797727510, −8.431662777003552260151574237361, −8.107254999974141425733612226723, −8.027820702389442688812467601575, −7.63991990852734577553267530836, −7.07858802468233348800829322538, −6.82722978419181667164956850862, −6.68144037078655771021164295697, −6.44053010502898545213009758947, −6.42303801021836928459479527874, −6.25573032325991831145670480609, −5.78243200103484308714693569016, −5.08061973141520009686416015880, −4.92878379485521351559472110760, −4.77226171212387565280923462842, −4.60110141709492036974659436557, −3.80135711138864567527566904674, −3.76294563482214022757096069319, −3.49889497051532979730464336803, −3.29884724948060516704191743938, −2.36332118847499301302913558219, −2.29880230993597420210544542444, −2.11554593785227139652540245546, −0.52493485583485650390044569946,
0.52493485583485650390044569946, 2.11554593785227139652540245546, 2.29880230993597420210544542444, 2.36332118847499301302913558219, 3.29884724948060516704191743938, 3.49889497051532979730464336803, 3.76294563482214022757096069319, 3.80135711138864567527566904674, 4.60110141709492036974659436557, 4.77226171212387565280923462842, 4.92878379485521351559472110760, 5.08061973141520009686416015880, 5.78243200103484308714693569016, 6.25573032325991831145670480609, 6.42303801021836928459479527874, 6.44053010502898545213009758947, 6.68144037078655771021164295697, 6.82722978419181667164956850862, 7.07858802468233348800829322538, 7.63991990852734577553267530836, 8.027820702389442688812467601575, 8.107254999974141425733612226723, 8.431662777003552260151574237361, 9.021931076140402793235797727510, 9.032662994130590356893921312244
