| L(s) = 1 | + 2-s + 3·3-s + 4-s + 3·5-s + 3·6-s + 6·9-s + 3·10-s − 3·11-s + 3·12-s + 8·13-s + 9·15-s − 16-s + 3·17-s + 6·18-s − 5·19-s + 3·20-s − 3·22-s − 23-s − 9·25-s + 8·26-s + 10·27-s + 2·29-s + 9·30-s − 31-s + 2·32-s − 9·33-s + 3·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.73·3-s + 1/2·4-s + 1.34·5-s + 1.22·6-s + 2·9-s + 0.948·10-s − 0.904·11-s + 0.866·12-s + 2.21·13-s + 2.32·15-s − 1/4·16-s + 0.727·17-s + 1.41·18-s − 1.14·19-s + 0.670·20-s − 0.639·22-s − 0.208·23-s − 9/5·25-s + 1.56·26-s + 1.92·27-s + 0.371·29-s + 1.64·30-s − 0.179·31-s + 0.353·32-s − 1.56·33-s + 0.514·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 7^{6} \cdot 41^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 7^{6} \cdot 41^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(29.80049528\) |
| \(L(\frac12)\) |
\(\approx\) |
\(29.80049528\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 7 | | \( 1 \) |
| 41 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 - T + T^{3} - p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{3} \) |
| 11 | $S_4\times C_2$ | \( 1 + 3 T + 19 T^{2} + 51 T^{3} + 19 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 8 T + 54 T^{2} - 213 T^{3} + 54 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 3 T + 7 T^{2} - 55 T^{3} + 7 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 5 T + 55 T^{2} + 189 T^{3} + 55 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + T + 21 T^{2} - 71 T^{3} + 21 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 2 T + 52 T^{2} - 161 T^{3} + 52 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + T + 25 T^{2} + 255 T^{3} + 25 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 16 T + 160 T^{2} - 29 p T^{3} + 160 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 8 T + 85 T^{2} - 376 T^{3} + 85 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 2 T + 2 p T^{2} - 257 T^{3} + 2 p^{2} T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 13 T + 199 T^{2} - 1369 T^{3} + 199 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 11 T + 181 T^{2} - 1283 T^{3} + 181 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 8 T + 138 T^{2} - 681 T^{3} + 138 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 17 T + 281 T^{2} + 2387 T^{3} + 281 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 29 T + 428 T^{2} - 4233 T^{3} + 428 p T^{4} - 29 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 22 T + 370 T^{2} - 3519 T^{3} + 370 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 24 T + 412 T^{2} + 4169 T^{3} + 412 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 9 T - 41 T^{2} + 1469 T^{3} - 41 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 4 T + 204 T^{2} + 839 T^{3} + 204 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 15 T + 138 T^{2} - 1263 T^{3} + 138 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.36824024708039911962406580469, −6.83459985149878170613781171074, −6.47650458438895829310034929561, −6.29303199960504774829252728219, −6.22763144529509422762793404796, −5.91348163601984603299704434977, −5.85778177905345949876537048136, −5.55155749432516606033743773023, −5.14723183678854017392906506818, −5.10526174567022233571579775368, −4.42758513224515788412158817139, −4.39458018930800914491975968937, −4.11039615784566442155073413647, −3.80784787441956688316940140899, −3.59741269773200738131393873276, −3.52654355486242392301079349896, −3.06174211996396323307549955448, −2.64537515632256261190716565875, −2.31997375119694202587499412486, −2.29444346669550692704697545835, −2.25131295006123011009485616706, −1.67235401106601636072438354426, −1.35861341769387647076698628567, −0.824849414721159007893695125748, −0.68244213074063501617941120045,
0.68244213074063501617941120045, 0.824849414721159007893695125748, 1.35861341769387647076698628567, 1.67235401106601636072438354426, 2.25131295006123011009485616706, 2.29444346669550692704697545835, 2.31997375119694202587499412486, 2.64537515632256261190716565875, 3.06174211996396323307549955448, 3.52654355486242392301079349896, 3.59741269773200738131393873276, 3.80784787441956688316940140899, 4.11039615784566442155073413647, 4.39458018930800914491975968937, 4.42758513224515788412158817139, 5.10526174567022233571579775368, 5.14723183678854017392906506818, 5.55155749432516606033743773023, 5.85778177905345949876537048136, 5.91348163601984603299704434977, 6.22763144529509422762793404796, 6.29303199960504774829252728219, 6.47650458438895829310034929561, 6.83459985149878170613781171074, 7.36824024708039911962406580469
