| L(s) = 1 | − 3·7-s − 5·9-s + 3·11-s − 13-s + 3·19-s − 7·23-s − 11·25-s − 27-s + 11·29-s − 6·31-s + 11·37-s − 5·41-s − 9·43-s + 4·47-s − 11·49-s − 4·53-s − 15·59-s − 4·61-s + 15·63-s − 8·67-s − 18·71-s + 6·73-s − 9·77-s − 4·79-s + 10·81-s − 21·83-s − 7·89-s + ⋯ |
| L(s) = 1 | − 1.13·7-s − 5/3·9-s + 0.904·11-s − 0.277·13-s + 0.688·19-s − 1.45·23-s − 2.19·25-s − 0.192·27-s + 2.04·29-s − 1.07·31-s + 1.80·37-s − 0.780·41-s − 1.37·43-s + 0.583·47-s − 1.57·49-s − 0.549·53-s − 1.95·59-s − 0.512·61-s + 1.88·63-s − 0.977·67-s − 2.13·71-s + 0.702·73-s − 1.02·77-s − 0.450·79-s + 10/9·81-s − 2.30·83-s − 0.741·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 11^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 11^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{3} \) |
| 19 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + 5 T^{2} + T^{3} + 5 p T^{4} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + 11 T^{2} - T^{3} + 11 p T^{4} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 3 T + 20 T^{2} + 40 T^{3} + 20 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + T + 20 T^{2} - 6 T^{3} + 20 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 35 T^{2} - 8 T^{3} + 35 p T^{4} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 7 T + 79 T^{2} + 318 T^{3} + 79 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 11 T + 112 T^{2} - 654 T^{3} + 112 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 6 T + 59 T^{2} + 11 p T^{3} + 59 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 11 T + 139 T^{2} - 830 T^{3} + 139 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 5 T + 46 T^{2} + 198 T^{3} + 46 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 9 T + 108 T^{2} + 556 T^{3} + 108 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 4 T + 97 T^{2} - 408 T^{3} + 97 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 4 T + 143 T^{2} + 368 T^{3} + 143 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 15 T + 209 T^{2} + 1786 T^{3} + 209 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 4 T + 163 T^{2} + 472 T^{3} + 163 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 8 T + 91 T^{2} + 881 T^{3} + 91 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 18 T + 317 T^{2} + 2749 T^{3} + 317 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 6 T + 215 T^{2} - 844 T^{3} + 215 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 4 T + 181 T^{2} + 736 T^{3} + 181 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 21 T + 320 T^{2} + 3500 T^{3} + 320 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 7 T + 69 T^{2} + 1098 T^{3} + 69 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 15 T + 323 T^{2} + 2926 T^{3} + 323 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.152840845837146822600080047763, −7.74756582781320630005518804731, −7.64471258429700450551257144468, −7.22387985488260394409944460787, −7.01025063476052120678470233586, −6.60745364654659298938248543098, −6.52575724874111789937293189175, −6.12067336988317441885015201964, −5.98883986714914390561665367550, −5.91459120260320057526695417012, −5.52664825105120138219851834994, −5.41146925637822186965134541099, −4.84473344504763201830955369903, −4.58541389226076648713913708633, −4.39107471273844852053366895686, −4.11811638777449084684066619801, −3.58819790716930589746762138130, −3.47185110883522129634359445653, −3.33300631734988120194837460608, −2.72046684190849323777623572499, −2.61320856228651096521296741346, −2.54034385163488225300900809153, −1.67739542852901017710678863099, −1.49405933279334037037371524387, −1.25233195549589881103109462143, 0, 0, 0,
1.25233195549589881103109462143, 1.49405933279334037037371524387, 1.67739542852901017710678863099, 2.54034385163488225300900809153, 2.61320856228651096521296741346, 2.72046684190849323777623572499, 3.33300631734988120194837460608, 3.47185110883522129634359445653, 3.58819790716930589746762138130, 4.11811638777449084684066619801, 4.39107471273844852053366895686, 4.58541389226076648713913708633, 4.84473344504763201830955369903, 5.41146925637822186965134541099, 5.52664825105120138219851834994, 5.91459120260320057526695417012, 5.98883986714914390561665367550, 6.12067336988317441885015201964, 6.52575724874111789937293189175, 6.60745364654659298938248543098, 7.01025063476052120678470233586, 7.22387985488260394409944460787, 7.64471258429700450551257144468, 7.74756582781320630005518804731, 8.152840845837146822600080047763
