| L(s) = 1 | + 3·7-s − 8-s + 18·11-s − 8·27-s − 12·31-s − 12·37-s + 24·49-s − 3·56-s + 54·77-s + 18·83-s − 18·88-s + 42·103-s − 27·107-s − 36·113-s + 141·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 1.13·7-s − 0.353·8-s + 5.42·11-s − 1.53·27-s − 2.15·31-s − 1.97·37-s + 24/7·49-s − 0.400·56-s + 6.15·77-s + 1.97·83-s − 1.91·88-s + 4.13·103-s − 2.61·107-s − 3.38·113-s + 12.8·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.670593731\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.670593731\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T^{3} + T^{6} \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 8 T^{3} + 37 T^{6} + 8 p^{3} T^{9} + p^{6} T^{12} \) |
| 5 | \( 1 - p^{3} T^{6} + p^{6} T^{12} \) |
| 7 | \( ( 1 - 5 T + p T^{2} )^{3}( 1 + 4 T + p T^{2} )^{3} \) |
| 11 | \( ( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{3} \) |
| 13 | \( ( 1 - 19 T^{3} + p^{3} T^{6} )( 1 + 89 T^{3} + p^{3} T^{6} ) \) |
| 17 | \( 1 - 126 T^{3} + 10963 T^{6} - 126 p^{3} T^{9} + p^{6} T^{12} \) |
| 23 | \( 1 - 180 T^{3} + 20233 T^{6} - 180 p^{3} T^{9} + p^{6} T^{12} \) |
| 29 | \( 1 + 54 T^{3} - 21473 T^{6} + 54 p^{3} T^{9} + p^{6} T^{12} \) |
| 31 | \( ( 1 - 7 T + p T^{2} )^{3}( 1 + 11 T + p T^{2} )^{3} \) |
| 37 | \( ( 1 + 2 T + p T^{2} )^{6} \) |
| 41 | \( 1 - p^{3} T^{6} + p^{6} T^{12} \) |
| 43 | \( ( 1 - 449 T^{3} + p^{3} T^{6} )( 1 - 71 T^{3} + p^{3} T^{6} ) \) |
| 47 | \( 1 - p^{3} T^{6} + p^{6} T^{12} \) |
| 53 | \( 1 - 450 T^{3} + 53623 T^{6} - 450 p^{3} T^{9} + p^{6} T^{12} \) |
| 59 | \( 1 + 864 T^{3} + 541117 T^{6} + 864 p^{3} T^{9} + p^{6} T^{12} \) |
| 61 | \( 1 + 830 T^{3} + 461919 T^{6} + 830 p^{3} T^{9} + p^{6} T^{12} \) |
| 67 | \( ( 1 - 127 T^{3} + p^{3} T^{6} )( 1 + 1007 T^{3} + p^{3} T^{6} ) \) |
| 71 | \( 1 - 1062 T^{3} + 769933 T^{6} - 1062 p^{3} T^{9} + p^{6} T^{12} \) |
| 73 | \( ( 1 + 271 T^{3} + p^{3} T^{6} )( 1 + 919 T^{3} + p^{3} T^{6} ) \) |
| 79 | \( 1 - 1370 T^{3} + 1383861 T^{6} - 1370 p^{3} T^{9} + p^{6} T^{12} \) |
| 83 | \( ( 1 - 6 T - 47 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{3} \) |
| 89 | \( 1 - 1476 T^{3} + 1473607 T^{6} - 1476 p^{3} T^{9} + p^{6} T^{12} \) |
| 97 | \( 1 - 1910 T^{3} + 2735427 T^{6} - 1910 p^{3} T^{9} + p^{6} T^{12} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.58015507663847693662402630872, −5.38940808284644924323253182810, −5.29064296469945187160141977663, −5.11152448676923494957027906561, −4.73702389317742243238283135910, −4.72018300970412051946087284715, −4.50349514610211168859969136946, −4.27771942799594674446980480582, −3.95902906327944848217397195939, −3.92285761863944884069613880240, −3.77875520861388078911615037336, −3.76368272517226243596758619668, −3.71727111308198691168348134962, −3.30213451960574396325027067848, −3.25043477833002597993343221795, −2.72730879810629324886427028574, −2.66486529464486256595206532921, −2.16816774011449608256489919894, −1.94784069035861619578230033458, −1.93290542123712423715834431658, −1.62761660157931603573099112697, −1.32619934332249299742012475508, −1.30911764998984673611734410859, −0.903095117112165033300385754387, −0.44613405403961795590339865329,
0.44613405403961795590339865329, 0.903095117112165033300385754387, 1.30911764998984673611734410859, 1.32619934332249299742012475508, 1.62761660157931603573099112697, 1.93290542123712423715834431658, 1.94784069035861619578230033458, 2.16816774011449608256489919894, 2.66486529464486256595206532921, 2.72730879810629324886427028574, 3.25043477833002597993343221795, 3.30213451960574396325027067848, 3.71727111308198691168348134962, 3.76368272517226243596758619668, 3.77875520861388078911615037336, 3.92285761863944884069613880240, 3.95902906327944848217397195939, 4.27771942799594674446980480582, 4.50349514610211168859969136946, 4.72018300970412051946087284715, 4.73702389317742243238283135910, 5.11152448676923494957027906561, 5.29064296469945187160141977663, 5.38940808284644924323253182810, 5.58015507663847693662402630872
Plot not available for L-functions of degree greater than 10.
