| L(s) = 1 | − 8·7-s − 32·13-s + 32·19-s + 72·25-s + 120·31-s − 88·37-s + 144·43-s + 36·49-s − 200·61-s + 112·67-s + 272·73-s + 488·79-s + 256·91-s + 160·97-s + 40·103-s − 192·109-s + 372·121-s + 127-s + 131-s − 256·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 8/7·7-s − 2.46·13-s + 1.68·19-s + 2.87·25-s + 3.87·31-s − 2.37·37-s + 3.34·43-s + 0.734·49-s − 3.27·61-s + 1.67·67-s + 3.72·73-s + 6.17·79-s + 2.81·91-s + 1.64·97-s + 0.388·103-s − 1.76·109-s + 3.07·121-s + 0.00787·127-s + 0.00763·131-s − 1.92·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(5.028939093\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.028939093\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 72 T^{2} + 98 p^{2} T^{4} - 72 p^{4} T^{6} + p^{8} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 + 4 T + 6 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 372 T^{2} + 62342 T^{4} - 372 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 16 T + 186 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 288 T^{2} + 184322 T^{4} - 288 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 16 T + 690 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 858 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 2152 T^{2} + 2530002 T^{4} - 2152 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 60 T + 2726 T^{2} - 60 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 44 T + 2838 T^{2} + 44 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 2560 T^{2} + 3056322 T^{4} - 2560 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 72 T + 3458 T^{2} - 72 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 6900 T^{2} + 21047462 T^{4} - 6900 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 7720 T^{2} + 28930962 T^{4} - 7720 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 1500 T^{2} + 2678822 T^{4} + 1500 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 100 T + 8406 T^{2} + 100 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 - 56 T + 9666 T^{2} - 56 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 5044 T^{2} + 16873062 T^{4} - 5044 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 136 T + 14898 T^{2} - 136 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 244 T + 26502 T^{2} - 244 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 1484 T^{2} - 19009338 T^{4} + 1484 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 11712 T^{2} + 138025922 T^{4} - 11712 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 80 T + 4194 T^{2} - 80 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−6.88961183828961391015109324318, −6.50223555695454273667829709796, −6.32670867689445088743032657016, −6.27769106380536491595246363123, −6.14208382147177271435000910262, −5.49778010711917415983575565184, −5.47678830966665186965807040883, −5.11811899401695851129424763336, −4.93282626490535246798707245544, −4.79847217933596630385796533873, −4.65400458477585401415392265607, −4.43999388072695122446673076133, −4.05026804549170209153675096445, −3.53875495838113256121120469402, −3.35038757233585790783861980824, −3.24268340789671488726901523034, −3.10426346327897135685863972591, −2.50647276578129634348582102298, −2.48327639166800387397039894368, −2.26671852764699627919093088516, −1.97948683099674901851727037853, −1.12494548331399667867745763690, −0.841463943587056048169589226301, −0.804914850890778932411711137567, −0.40491247981270283916694061408,
0.40491247981270283916694061408, 0.804914850890778932411711137567, 0.841463943587056048169589226301, 1.12494548331399667867745763690, 1.97948683099674901851727037853, 2.26671852764699627919093088516, 2.48327639166800387397039894368, 2.50647276578129634348582102298, 3.10426346327897135685863972591, 3.24268340789671488726901523034, 3.35038757233585790783861980824, 3.53875495838113256121120469402, 4.05026804549170209153675096445, 4.43999388072695122446673076133, 4.65400458477585401415392265607, 4.79847217933596630385796533873, 4.93282626490535246798707245544, 5.11811899401695851129424763336, 5.47678830966665186965807040883, 5.49778010711917415983575565184, 6.14208382147177271435000910262, 6.27769106380536491595246363123, 6.32670867689445088743032657016, 6.50223555695454273667829709796, 6.88961183828961391015109324318
