| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 3^{3} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 3^{3} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.26147978\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.26147978\) |
| \(L(\frac{3}{2})\) |
|
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
−7.54060323735456651837810979878, −7.07415820956782095813385149489, −6.82593107590799135907221048444, −6.80093750535903022582386440296, −6.37984307927286694809904602665, −6.31064040280517815199869036234, −5.96012465133648246777793467390, −5.51646067411035830483502473946, −5.40182101631823195871611059999, −5.00199051123414235311694269462, −4.71863203383945951025678417052, −4.52053108245893864886756175343, −4.29826280691441894826595607303, −3.89089382393204731167696561924, −3.73487988306114784297201057831, −3.53535953043013448970352082283, −3.10897084028652213437892636799, −2.79601212502706247141468895745, −2.68008481434923792509129963594, −2.32017412104279262723917433897, −2.02081170243445436849726652818, −1.76413834142388224342774442761, −1.05453466129633628542132417669, −1.03890672942108314003403536911, −0.52916244062833918146348448157,
0.52916244062833918146348448157, 1.03890672942108314003403536911, 1.05453466129633628542132417669, 1.76413834142388224342774442761, 2.02081170243445436849726652818, 2.32017412104279262723917433897, 2.68008481434923792509129963594, 2.79601212502706247141468895745, 3.10897084028652213437892636799, 3.53535953043013448970352082283, 3.73487988306114784297201057831, 3.89089382393204731167696561924, 4.29826280691441894826595607303, 4.52053108245893864886756175343, 4.71863203383945951025678417052, 5.00199051123414235311694269462, 5.40182101631823195871611059999, 5.51646067411035830483502473946, 5.96012465133648246777793467390, 6.31064040280517815199869036234, 6.37984307927286694809904602665, 6.80093750535903022582386440296, 6.82593107590799135907221048444, 7.07415820956782095813385149489, 7.54060323735456651837810979878
