| L(s) = 1 | + 2·4-s + 4·7-s − 2·13-s − 2·19-s − 4·25-s + 8·28-s − 2·31-s + 16·37-s + 22·43-s − 2·49-s − 4·52-s + 10·61-s − 8·64-s − 14·67-s + 22·73-s − 4·76-s − 14·79-s − 8·91-s − 14·97-s − 8·100-s − 14·103-s − 2·109-s − 16·121-s − 4·124-s + 127-s + 131-s − 8·133-s + ⋯ |
| L(s) = 1 | + 4-s + 1.51·7-s − 0.554·13-s − 0.458·19-s − 4/5·25-s + 1.51·28-s − 0.359·31-s + 2.63·37-s + 3.35·43-s − 2/7·49-s − 0.554·52-s + 1.28·61-s − 64-s − 1.71·67-s + 2.57·73-s − 0.458·76-s − 1.57·79-s − 0.838·91-s − 1.42·97-s − 4/5·100-s − 1.37·103-s − 0.191·109-s − 1.45·121-s − 0.359·124-s + 0.0887·127-s + 0.0873·131-s − 0.693·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59049 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59049 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.018472583\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.018472583\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 52 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 112 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 88 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
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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
−12.18863131069501005241351840304, −11.71630489752962138017706271769, −11.36341435559568201719934172804, −10.98957259426085812164359937729, −10.72633054857364640549729516398, −10.01624557525827670252698459237, −9.373503120959489145087639531418, −9.086887255792082165042347510051, −8.195021880370970742472551747703, −7.79886961415182704033617479477, −7.60162825894856137206908174096, −6.88522256932747131908927626690, −6.28792838733343724695047154582, −5.74897891895457914005886933698, −5.14855857250750191167305147161, −4.35591997396063863904045940049, −4.05301410065910786040085000782, −2.65079034130041710876826775177, −2.34752347581144079652748859470, −1.35764139016953185842768122715,
1.35764139016953185842768122715, 2.34752347581144079652748859470, 2.65079034130041710876826775177, 4.05301410065910786040085000782, 4.35591997396063863904045940049, 5.14855857250750191167305147161, 5.74897891895457914005886933698, 6.28792838733343724695047154582, 6.88522256932747131908927626690, 7.60162825894856137206908174096, 7.79886961415182704033617479477, 8.195021880370970742472551747703, 9.086887255792082165042347510051, 9.373503120959489145087639531418, 10.01624557525827670252698459237, 10.72633054857364640549729516398, 10.98957259426085812164359937729, 11.36341435559568201719934172804, 11.71630489752962138017706271769, 12.18863131069501005241351840304
