| L(s) = 1 | + 2·3-s − 4-s − 7-s + 3·9-s − 2·12-s − 13-s − 2·19-s − 2·21-s − 25-s + 4·27-s + 28-s − 2·31-s − 3·36-s + 37-s − 2·39-s + 43-s + 52-s − 4·57-s − 2·61-s − 3·63-s + 64-s + 4·67-s + 73-s − 2·75-s + 2·76-s − 2·79-s + 5·81-s + ⋯ |
| L(s) = 1 | + 2·3-s − 4-s − 7-s + 3·9-s − 2·12-s − 13-s − 2·19-s − 2·21-s − 25-s + 4·27-s + 28-s − 2·31-s − 3·36-s + 37-s − 2·39-s + 43-s + 52-s − 4·57-s − 2·61-s − 3·63-s + 64-s + 4·67-s + 73-s − 2·75-s + 2·76-s − 2·79-s + 5·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7598292979\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7598292979\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| 13 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 67 | $C_1$ | \( ( 1 - T )^{4} \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.53134817347494461545917938812, −12.47280104986919417425236328904, −11.28312144833772637619063067850, −10.85357918957215016724031437838, −10.00632972835089425287224747310, −9.927840296809085645880369734637, −9.345112294070437519236864962480, −9.167265071928771324305868643822, −8.554337223331167530883043376707, −8.269488523470644085255250662983, −7.51989052799530073639003036007, −7.27257749214163409643245943034, −6.56356817623343802398847747667, −5.98868748900753155231223910514, −4.94176371987835500257443711914, −4.43874307070481477132483183021, −3.85365541895090556900335790972, −3.49716115974458044202166729864, −2.46618336311894048396802786649, −2.08137434854167615783808386896,
2.08137434854167615783808386896, 2.46618336311894048396802786649, 3.49716115974458044202166729864, 3.85365541895090556900335790972, 4.43874307070481477132483183021, 4.94176371987835500257443711914, 5.98868748900753155231223910514, 6.56356817623343802398847747667, 7.27257749214163409643245943034, 7.51989052799530073639003036007, 8.269488523470644085255250662983, 8.554337223331167530883043376707, 9.167265071928771324305868643822, 9.345112294070437519236864962480, 9.927840296809085645880369734637, 10.00632972835089425287224747310, 10.85357918957215016724031437838, 11.28312144833772637619063067850, 12.47280104986919417425236328904, 12.53134817347494461545917938812
