| L(s) = 1 | − 4·3-s − 3·5-s − 2·7-s + 5·9-s − 10·11-s + 4·13-s + 12·15-s + 6·17-s + 3·19-s + 8·21-s + 2·23-s + 6·25-s + 2·27-s + 14·29-s + 2·31-s + 40·33-s + 6·35-s − 2·37-s − 16·39-s + 4·41-s − 14·43-s − 15·45-s − 2·47-s − 9·49-s − 24·51-s + 14·53-s + 30·55-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 1.34·5-s − 0.755·7-s + 5/3·9-s − 3.01·11-s + 1.10·13-s + 3.09·15-s + 1.45·17-s + 0.688·19-s + 1.74·21-s + 0.417·23-s + 6/5·25-s + 0.384·27-s + 2.59·29-s + 0.359·31-s + 6.96·33-s + 1.01·35-s − 0.328·37-s − 2.56·39-s + 0.624·41-s − 2.13·43-s − 2.23·45-s − 0.291·47-s − 9/7·49-s − 3.36·51-s + 1.92·53-s + 4.04·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) |
| 19 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + 4 T + 11 T^{2} + 22 T^{3} + 11 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 2 T + 13 T^{2} + 32 T^{3} + 13 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 10 T + 61 T^{2} + 240 T^{3} + 61 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 4 T + 23 T^{2} - 114 T^{3} + 23 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) |
| 23 | $S_4\times C_2$ | \( 1 - 2 T + 61 T^{2} - 96 T^{3} + 61 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $D_{6}$ | \( 1 - 14 T + 91 T^{2} - 468 T^{3} + 91 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 2 T + 41 T^{2} + 60 T^{3} + 41 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 2 T + 91 T^{2} + 98 T^{3} + 91 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 4 T + 67 T^{2} - 408 T^{3} + 67 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 14 T + 37 T^{2} - 184 T^{3} + 37 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 2 T + 133 T^{2} + 192 T^{3} + 133 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 14 T + 203 T^{2} - 1450 T^{3} + 203 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 10 T + 189 T^{2} + 1140 T^{3} + 189 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 10 T + 31 T^{2} - 224 T^{3} + 31 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 6 T + 143 T^{2} - 586 T^{3} + 143 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 2 T + 161 T^{2} - 100 T^{3} + 161 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 10 T + 191 T^{2} - 1452 T^{3} + 191 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 14 T + 169 T^{2} - 1100 T^{3} + 169 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 2 T + 101 T^{2} + 464 T^{3} + 101 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 2 T + 207 T^{2} - 156 T^{3} + 207 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 10 T + 259 T^{2} - 1894 T^{3} + 259 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.57341621249441350487245104691, −6.99819329057257606616194639788, −6.98100430878597522383864785830, −6.81270428850960777155687697103, −6.38031310062026990225985139029, −6.24522792206782373099169042844, −6.17951619778166335729804448237, −5.70515092866181007260563194121, −5.44295474024535211009573834635, −5.41268512723719867594670927371, −5.03696367395824352363636136524, −5.01505391906774734952464752012, −4.85811350275164414858076884842, −4.41937508413274422834326426203, −4.15683079864013331790346314076, −3.74462723324323709367970730428, −3.31466219632723102692444008179, −3.24631044828590750152350328440, −3.10759920936491044041559212853, −2.79190976770020417797210846418, −2.32746354685020143711793545997, −2.18127591706728015314077057080, −1.14165833108343166890997115044, −1.11880242434552781746461996068, −0.907692605326591438652290872169, 0, 0, 0,
0.907692605326591438652290872169, 1.11880242434552781746461996068, 1.14165833108343166890997115044, 2.18127591706728015314077057080, 2.32746354685020143711793545997, 2.79190976770020417797210846418, 3.10759920936491044041559212853, 3.24631044828590750152350328440, 3.31466219632723102692444008179, 3.74462723324323709367970730428, 4.15683079864013331790346314076, 4.41937508413274422834326426203, 4.85811350275164414858076884842, 5.01505391906774734952464752012, 5.03696367395824352363636136524, 5.41268512723719867594670927371, 5.44295474024535211009573834635, 5.70515092866181007260563194121, 6.17951619778166335729804448237, 6.24522792206782373099169042844, 6.38031310062026990225985139029, 6.81270428850960777155687697103, 6.98100430878597522383864785830, 6.99819329057257606616194639788, 7.57341621249441350487245104691
