| L(s) = 1 | + 3·3-s + 6·9-s + 3·13-s + 6·17-s + 3·19-s + 6·23-s − 6·25-s + 10·27-s − 12·29-s − 3·31-s − 3·37-s + 9·39-s − 6·41-s + 15·43-s − 12·47-s + 18·51-s − 6·53-s + 9·57-s + 12·59-s + 18·61-s + 9·67-s + 18·69-s + 33·73-s − 18·75-s + 27·79-s + 15·81-s + 18·83-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 2·9-s + 0.832·13-s + 1.45·17-s + 0.688·19-s + 1.25·23-s − 6/5·25-s + 1.92·27-s − 2.22·29-s − 0.538·31-s − 0.493·37-s + 1.44·39-s − 0.937·41-s + 2.28·43-s − 1.75·47-s + 2.52·51-s − 0.824·53-s + 1.19·57-s + 1.56·59-s + 2.30·61-s + 1.09·67-s + 2.16·69-s + 3.86·73-s − 2.07·75-s + 3.03·79-s + 5/3·81-s + 1.97·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{3} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{3} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(16.57652455\) |
| \(L(\frac12)\) |
\(\approx\) |
\(16.57652455\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 7 | | \( 1 \) |
| good | 5 | $S_4\times C_2$ | \( 1 + 6 T^{2} - 4 T^{3} + 6 p T^{4} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 6 T^{2} + 38 T^{3} + 6 p T^{4} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 3 T + 3 T^{2} + 34 T^{3} + 3 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 6 T + 27 T^{2} - 108 T^{3} + 27 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 3 T + 21 T^{2} - 2 T^{3} + 21 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 6 T + 45 T^{2} - 180 T^{3} + 45 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 12 T + 126 T^{2} + 728 T^{3} + 126 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 3 T + 72 T^{2} + 139 T^{3} + 72 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 3 T + 27 T^{2} - 146 T^{3} + 27 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 6 T + 27 T^{2} - 20 T^{3} + 27 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 15 T + 177 T^{2} - 1318 T^{3} + 177 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 12 T + 153 T^{2} + 1016 T^{3} + 153 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 6 T + 78 T^{2} + 802 T^{3} + 78 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 12 T + 198 T^{2} - 1334 T^{3} + 198 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 18 T + 195 T^{2} - 1644 T^{3} + 195 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 9 T + 189 T^{2} - 1042 T^{3} + 189 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 177 T^{2} + 32 T^{3} + 177 p T^{4} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 33 T + 555 T^{2} - 5814 T^{3} + 555 p T^{4} - 33 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 27 T + 432 T^{2} - 4619 T^{3} + 432 p T^{4} - 27 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 18 T + 234 T^{2} - 2040 T^{3} + 234 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 12 T + 279 T^{2} + 2024 T^{3} + 279 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 264 T^{2} + 38 T^{3} + 264 p T^{4} + p^{3} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.79702285021151383294216490740, −6.69970388522357735383395338939, −6.41704899927950004006749425362, −6.26790084062295478720921760878, −5.75290817132326567045518145549, −5.59847989632087425520188919343, −5.39370290489604121782443616198, −5.04024394122939073898479909759, −5.03872647249113258508282305961, −4.93948795041137237196621098882, −4.04619748574095065356313440229, −4.00472874836550369570632017503, −3.96185821880639667051699278126, −3.58388091753829931177248917921, −3.52359300724483573880395732940, −3.31506309007018841512803721446, −2.81836846825119124491307093744, −2.75703386274897823236741822037, −2.33867185541997348910026149245, −1.89445602034363284163293104649, −1.79623899672329126785494214777, −1.74520556254605719345010176049, −1.00842652589509993179592572831, −0.75899106749662383790908615577, −0.57197262914858841918908837929,
0.57197262914858841918908837929, 0.75899106749662383790908615577, 1.00842652589509993179592572831, 1.74520556254605719345010176049, 1.79623899672329126785494214777, 1.89445602034363284163293104649, 2.33867185541997348910026149245, 2.75703386274897823236741822037, 2.81836846825119124491307093744, 3.31506309007018841512803721446, 3.52359300724483573880395732940, 3.58388091753829931177248917921, 3.96185821880639667051699278126, 4.00472874836550369570632017503, 4.04619748574095065356313440229, 4.93948795041137237196621098882, 5.03872647249113258508282305961, 5.04024394122939073898479909759, 5.39370290489604121782443616198, 5.59847989632087425520188919343, 5.75290817132326567045518145549, 6.26790084062295478720921760878, 6.41704899927950004006749425362, 6.69970388522357735383395338939, 6.79702285021151383294216490740
