| L(s) = 1 | − 2-s − 3·3-s − 3·4-s + 3·5-s + 3·6-s + 2·7-s + 4·8-s + 6·9-s − 3·10-s − 3·11-s + 9·12-s − 2·14-s − 9·15-s + 3·16-s − 10·17-s − 6·18-s − 6·19-s − 9·20-s − 6·21-s + 3·22-s + 5·23-s − 12·24-s + 6·25-s − 10·27-s − 6·28-s + 29-s + 9·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s − 3/2·4-s + 1.34·5-s + 1.22·6-s + 0.755·7-s + 1.41·8-s + 2·9-s − 0.948·10-s − 0.904·11-s + 2.59·12-s − 0.534·14-s − 2.32·15-s + 3/4·16-s − 2.42·17-s − 1.41·18-s − 1.37·19-s − 2.01·20-s − 1.30·21-s + 0.639·22-s + 1.04·23-s − 2.44·24-s + 6/5·25-s − 1.92·27-s − 1.13·28-s + 0.185·29-s + 1.64·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) |
| 13 | | \( 1 \) |
| good | 2 | $A_4\times C_2$ | \( 1 + T + p^{2} T^{2} + 3 T^{3} + p^{3} T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 - 2 T + 20 T^{2} - 27 T^{3} + 20 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $C_2$ | \( ( 1 + T + p T^{2} )^{3} \) |
| 17 | $A_4\times C_2$ | \( 1 + 10 T + 75 T^{2} + 348 T^{3} + 75 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 + 6 T + 48 T^{2} + 187 T^{3} + 48 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 - 5 T + 75 T^{2} - 231 T^{3} + 75 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 - T + 43 T^{2} + 69 T^{3} + 43 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 - 3 T + 68 T^{2} - 215 T^{3} + 68 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 - 5 T + 89 T^{2} - 273 T^{3} + 89 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 + 22 T + 282 T^{2} + 2181 T^{3} + 282 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 + 16 T + 212 T^{2} + 1515 T^{3} + 212 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 - 12 T + 140 T^{2} - 1087 T^{3} + 140 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 - 5 T + 67 T^{2} - 447 T^{3} + 67 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 - 3 T + 89 T^{2} + 23 T^{3} + 89 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 10 T + 102 T^{2} + 787 T^{3} + 102 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 + 20 T + 318 T^{2} + 2849 T^{3} + 318 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 - 6 T + 162 T^{2} - 545 T^{3} + 162 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - 2 T + 36 T^{2} + 451 T^{3} + 36 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 2 T + 82 T^{2} + 539 T^{3} + 82 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 - 27 T + 401 T^{2} - 4105 T^{3} + 401 p T^{4} - 27 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 - 4 T + 186 T^{2} - 291 T^{3} + 186 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 - 9 T + 66 T^{2} + 495 T^{3} + 66 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.385021531681030757262889128798, −8.136709309712197690401334649658, −7.73571092440817019125064642193, −7.67911543210994216716026169044, −7.21383224542839236343697968962, −6.76395103304990348121644311631, −6.62428381059084084339818621672, −6.37336560752757877990376103268, −6.35523222557181913141604841870, −6.10822262304532670527591280914, −5.49238347787113414752154873342, −5.13278296632894632542990279075, −5.11958281189822046855103317155, −4.78706303183222893808563116355, −4.75901408992017927029275047645, −4.73147268119568970027292089316, −3.96625920283646726041518113664, −3.86300856561736973465135641827, −3.55505344744914323959901217104, −2.74953477905397220149311528273, −2.51294403224615397100309904774, −2.24782800739701030175330770637, −1.68788390320668760784852873942, −1.29868854289086909134817235358, −1.22994124126641937748704070121, 0, 0, 0,
1.22994124126641937748704070121, 1.29868854289086909134817235358, 1.68788390320668760784852873942, 2.24782800739701030175330770637, 2.51294403224615397100309904774, 2.74953477905397220149311528273, 3.55505344744914323959901217104, 3.86300856561736973465135641827, 3.96625920283646726041518113664, 4.73147268119568970027292089316, 4.75901408992017927029275047645, 4.78706303183222893808563116355, 5.11958281189822046855103317155, 5.13278296632894632542990279075, 5.49238347787113414752154873342, 6.10822262304532670527591280914, 6.35523222557181913141604841870, 6.37336560752757877990376103268, 6.62428381059084084339818621672, 6.76395103304990348121644311631, 7.21383224542839236343697968962, 7.67911543210994216716026169044, 7.73571092440817019125064642193, 8.136709309712197690401334649658, 8.385021531681030757262889128798
