| L(s) = 1 | − 8-s + 3·11-s − 12·13-s − 6·17-s − 3·19-s − 9·23-s + 15·29-s + 15·31-s − 12·37-s − 12·41-s − 12·43-s + 12·47-s − 9·53-s − 12·59-s + 3·61-s − 7·64-s − 15·67-s − 18·71-s − 3·73-s + 3·79-s + 3·83-s − 3·88-s + 3·89-s + 12·97-s − 6·101-s + 15·103-s + 12·104-s + ⋯ |
| L(s) = 1 | − 0.353·8-s + 0.904·11-s − 3.32·13-s − 1.45·17-s − 0.688·19-s − 1.87·23-s + 2.78·29-s + 2.69·31-s − 1.97·37-s − 1.87·41-s − 1.82·43-s + 1.75·47-s − 1.23·53-s − 1.56·59-s + 0.384·61-s − 7/8·64-s − 1.83·67-s − 2.13·71-s − 0.351·73-s + 0.337·79-s + 0.329·83-s − 0.319·88-s + 0.317·89-s + 1.21·97-s − 0.597·101-s + 1.47·103-s + 1.17·104-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 2 | $D_{6}$ | \( 1 + T^{3} + p^{3} T^{6} \) |
| 7 | $D_{6}$ | \( 1 - 16 T^{3} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 3 T + 21 T^{2} - 50 T^{3} + 21 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{3} \) |
| 17 | $S_4\times C_2$ | \( 1 + 6 T + 15 T^{2} - 4 T^{3} + 15 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 9 T + 75 T^{2} + 362 T^{3} + 75 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{3} \) |
| 31 | $S_4\times C_2$ | \( 1 - 15 T + 153 T^{2} - 978 T^{3} + 153 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 12 T + 135 T^{2} + 864 T^{3} + 135 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 12 T + 156 T^{2} + 990 T^{3} + 156 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{3} \) |
| 47 | $S_4\times C_2$ | \( 1 - 12 T + 129 T^{2} - 936 T^{3} + 129 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 9 T + 90 T^{2} + 757 T^{3} + 90 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 12 T + 168 T^{2} + 1096 T^{3} + 168 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 3 T + 126 T^{2} - 323 T^{3} + 126 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 15 T + 261 T^{2} + 2062 T^{3} + 261 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 18 T + 264 T^{2} + 2274 T^{3} + 264 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 3 T + 126 T^{2} + 407 T^{3} + 126 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 3 T + 51 T^{2} + 146 T^{3} + 51 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 3 T + 237 T^{2} - 486 T^{3} + 237 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 3 T + 186 T^{2} - 579 T^{3} + 186 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 12 T + 219 T^{2} - 1424 T^{3} + 219 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.84373162863912574932877542224, −7.48857453675460329811108663931, −7.28485447015335395290735934766, −6.96460348749361390818662512901, −6.82845868755271393030182908296, −6.56941720787635864575756749427, −6.33323484204109720734789561393, −6.18252882445352735209715341359, −6.00836693668124549668877335706, −5.56872325025590433515194561882, −4.94453168051928347778193564201, −4.93946596039844254161023269146, −4.88264579975966337625541967547, −4.51723115005951413555856300776, −4.40292486845125305721525921109, −4.12740026476696786251403039333, −3.51071478533013687157880070424, −3.48586444181951134127886006135, −2.83365591130468668665283005068, −2.71460639281247497168371872215, −2.62547216699874871537690867155, −2.06063082697105704858324256726, −1.98286325309074832804517608019, −1.37483772949975213331872288713, −1.15801004697462128474427604526, 0, 0, 0,
1.15801004697462128474427604526, 1.37483772949975213331872288713, 1.98286325309074832804517608019, 2.06063082697105704858324256726, 2.62547216699874871537690867155, 2.71460639281247497168371872215, 2.83365591130468668665283005068, 3.48586444181951134127886006135, 3.51071478533013687157880070424, 4.12740026476696786251403039333, 4.40292486845125305721525921109, 4.51723115005951413555856300776, 4.88264579975966337625541967547, 4.93946596039844254161023269146, 4.94453168051928347778193564201, 5.56872325025590433515194561882, 6.00836693668124549668877335706, 6.18252882445352735209715341359, 6.33323484204109720734789561393, 6.56941720787635864575756749427, 6.82845868755271393030182908296, 6.96460348749361390818662512901, 7.28485447015335395290735934766, 7.48857453675460329811108663931, 7.84373162863912574932877542224
