| L(s) = 1 | − 3·2-s − 3·3-s + 6·4-s − 3·5-s + 9·6-s − 7-s − 10·8-s + 6·9-s + 9·10-s + 5·11-s − 18·12-s + 3·13-s + 3·14-s + 9·15-s + 15·16-s + 7·17-s − 18·18-s + 3·19-s − 18·20-s + 3·21-s − 15·22-s + 11·23-s + 30·24-s − 4·25-s − 9·26-s − 10·27-s − 6·28-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.73·3-s + 3·4-s − 1.34·5-s + 3.67·6-s − 0.377·7-s − 3.53·8-s + 2·9-s + 2.84·10-s + 1.50·11-s − 5.19·12-s + 0.832·13-s + 0.801·14-s + 2.32·15-s + 15/4·16-s + 1.69·17-s − 4.24·18-s + 0.688·19-s − 4.02·20-s + 0.654·21-s − 3.19·22-s + 2.29·23-s + 6.12·24-s − 4/5·25-s − 1.76·26-s − 1.92·27-s − 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 19^{3} \cdot 53^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 19^{3} \cdot 53^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9114036630\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9114036630\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 3 | $C_1$ | \( ( 1 + T )^{3} \) |
| 19 | $C_1$ | \( ( 1 - T )^{3} \) |
| 53 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 5 | $S_4\times C_2$ | \( 1 + 3 T + 13 T^{2} + 23 T^{3} + 13 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + T + 11 T^{2} + 23 T^{3} + 11 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 5 T + 3 p T^{2} - 101 T^{3} + 3 p^{2} T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 3 T + 37 T^{2} - 71 T^{3} + 37 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 7 T + 57 T^{2} - 239 T^{3} + 57 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 11 T + 87 T^{2} - 439 T^{3} + 87 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 2 T + 10 T^{2} + 11 T^{3} + 10 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 4 T + 65 T^{2} - 208 T^{3} + 65 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 11 T + 143 T^{2} - 833 T^{3} + 143 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 14 T + 162 T^{2} - 1103 T^{3} + 162 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 11 T + 147 T^{2} + 879 T^{3} + 147 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 11 T + 164 T^{2} + 1007 T^{3} + 164 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 6 T + 34 T^{2} + 225 T^{3} + 34 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 10 T + 138 T^{2} + 1071 T^{3} + 138 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 19 T + 297 T^{2} - 2609 T^{3} + 297 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 16 T + 281 T^{2} - 2344 T^{3} + 281 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 9 T + 205 T^{2} - 1319 T^{3} + 205 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 21 T + 183 T^{2} + 1325 T^{3} + 183 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 15 T + 193 T^{2} - 1491 T^{3} + 193 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 4 T + 220 T^{2} + 659 T^{3} + 220 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 11 T + 251 T^{2} + 1613 T^{3} + 251 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.29796671364614912397572497093, −6.85334179343506122103541039265, −6.65594435316350701356686977856, −6.52376977298953710431400074018, −6.37255126390513906526923470171, −6.07317935469097713842482463199, −5.97761634426516133906539146170, −5.44994343887715437629214696224, −5.33481107082800819704502699178, −5.15258136012902906984237210945, −4.65782123624353588052094139944, −4.46350764381611434936297824160, −4.24181121667867412414242092575, −3.68004894194799848094417795996, −3.59688130107879071011398145871, −3.43829666412722359614515297414, −3.11580301221858800608677010678, −2.74055456985699345938949034621, −2.40344977479097595260028434341, −1.77964482059232141218335756163, −1.41643573621987703465073744020, −1.36895083408355173920955543660, −0.815872966209733362444652838303, −0.67604467294507715184735691463, −0.47307858427833738323096286378,
0.47307858427833738323096286378, 0.67604467294507715184735691463, 0.815872966209733362444652838303, 1.36895083408355173920955543660, 1.41643573621987703465073744020, 1.77964482059232141218335756163, 2.40344977479097595260028434341, 2.74055456985699345938949034621, 3.11580301221858800608677010678, 3.43829666412722359614515297414, 3.59688130107879071011398145871, 3.68004894194799848094417795996, 4.24181121667867412414242092575, 4.46350764381611434936297824160, 4.65782123624353588052094139944, 5.15258136012902906984237210945, 5.33481107082800819704502699178, 5.44994343887715437629214696224, 5.97761634426516133906539146170, 6.07317935469097713842482463199, 6.37255126390513906526923470171, 6.52376977298953710431400074018, 6.65594435316350701356686977856, 6.85334179343506122103541039265, 7.29796671364614912397572497093
