| L(s) = 1 | − 2·4-s − 5·9-s + 3·16-s + 10·36-s + 48·41-s − 4·49-s − 24·59-s − 12·64-s − 4·79-s + 9·81-s + 72·89-s − 48·101-s + 68·109-s + 74·121-s + 127-s + 131-s + 137-s + 139-s − 15·144-s + 149-s + 151-s + 157-s + 163-s − 96·164-s + 167-s − 2·169-s + 173-s + ⋯ |
| L(s) = 1 | − 4-s − 5/3·9-s + 3/4·16-s + 5/3·36-s + 7.49·41-s − 4/7·49-s − 3.12·59-s − 3/2·64-s − 0.450·79-s + 81-s + 7.63·89-s − 4.77·101-s + 6.51·109-s + 6.72·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5/4·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s − 7.49·164-s + 0.0773·167-s − 0.153·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.313070918\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.313070918\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 5 T^{2} + 16 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | \( 1 \) |
| 7 | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| good | 2 | \( ( 1 + T^{2} + p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 37 T^{2} + 576 T^{4} - 37 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 + T^{2} + 264 T^{4} + p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 47 T^{2} + 1056 T^{4} - 47 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 8 T + p T^{2} )^{4}( 1 + 8 T + p T^{2} )^{4} \) |
| 23 | \( ( 1 + 16 T^{2} - 66 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 97 T^{2} + 3960 T^{4} - 97 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 - 80 T^{2} + 4206 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 6 T + p T^{2} )^{8} \) |
| 43 | \( ( 1 - 104 T^{2} + 6270 T^{4} - 104 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 131 T^{2} + 8040 T^{4} - 131 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + 136 T^{2} + 9054 T^{4} + 136 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 6 T + 94 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - 14 T + p T^{2} )^{4}( 1 + 14 T + p T^{2} )^{4} \) |
| 67 | \( ( 1 - 200 T^{2} + 18846 T^{4} - 200 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 100 T^{2} + 4134 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 98 T^{2} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 + T + 150 T^{2} + p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 4 T^{2} - 5226 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 18 T + 226 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 + 265 T^{2} + 32736 T^{4} + 265 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.74954320705353896481787688276, −4.61266045500596025541020696534, −4.52709655438864827569252164949, −4.47397752291348499491910428551, −4.18399620478506669995141952196, −4.11287647505527501662565409763, −4.08953277855884655382921073590, −3.62335426485669827650163998968, −3.56820909678364148574777589644, −3.39105866018984720415073872461, −3.38662811378780551745848816007, −3.14550333890647839344860706442, −3.07940959258686842266018543238, −2.80670334735880229957353904350, −2.68253381951277483635724359338, −2.48592892310393052970360846779, −2.34834909882442939836985874476, −2.03996584249600337355091258269, −1.98196703965542509379873580277, −1.87915412785162903504348455049, −1.39317471071767236800489210447, −1.14983928922705511721362805448, −0.66797537338404652857394054483, −0.61256484650819105448868168073, −0.57372402697817544619201336355,
0.57372402697817544619201336355, 0.61256484650819105448868168073, 0.66797537338404652857394054483, 1.14983928922705511721362805448, 1.39317471071767236800489210447, 1.87915412785162903504348455049, 1.98196703965542509379873580277, 2.03996584249600337355091258269, 2.34834909882442939836985874476, 2.48592892310393052970360846779, 2.68253381951277483635724359338, 2.80670334735880229957353904350, 3.07940959258686842266018543238, 3.14550333890647839344860706442, 3.38662811378780551745848816007, 3.39105866018984720415073872461, 3.56820909678364148574777589644, 3.62335426485669827650163998968, 4.08953277855884655382921073590, 4.11287647505527501662565409763, 4.18399620478506669995141952196, 4.47397752291348499491910428551, 4.52709655438864827569252164949, 4.61266045500596025541020696534, 4.74954320705353896481787688276
Plot not available for L-functions of degree greater than 10.
