| L(s) = 1 | + 3·2-s − 9·3-s + 4-s + 4·5-s − 27·6-s − 22·7-s − 6·8-s + 45·9-s + 12·10-s − 36·11-s − 9·12-s − 3·13-s − 66·14-s − 36·15-s − 7·16-s + 38·17-s + 135·18-s − 10·19-s + 4·20-s + 198·21-s − 108·22-s + 54·23-s + 54·24-s + 35·25-s − 9·26-s − 162·27-s − 22·28-s + ⋯ |
| L(s) = 1 | + 3/2·2-s − 3·3-s + 1/4·4-s + 4/5·5-s − 9/2·6-s − 3.14·7-s − 3/4·8-s + 5·9-s + 6/5·10-s − 3.27·11-s − 3/4·12-s − 0.230·13-s − 4.71·14-s − 2.39·15-s − 0.437·16-s + 2.23·17-s + 15/2·18-s − 0.526·19-s + 1/5·20-s + 66/7·21-s − 4.90·22-s + 2.34·23-s + 9/4·24-s + 7/5·25-s − 0.346·26-s − 6·27-s − 0.785·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.02957177215\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02957177215\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( ( 1 + p T + p T^{2} )^{3} \) |
| 19 | \( 1 + 10 T - 249 T^{2} - 172 p T^{3} - 249 p^{2} T^{4} + 10 p^{4} T^{5} + p^{6} T^{6} \) |
| good | 2 | \( 1 - 3 T + p^{3} T^{2} - 15 T^{3} + 13 p T^{4} - 57 T^{5} + 71 T^{6} - 57 p^{2} T^{7} + 13 p^{5} T^{8} - 15 p^{6} T^{9} + p^{11} T^{10} - 3 p^{10} T^{11} + p^{12} T^{12} \) |
| 5 | \( 1 - 4 T - 19 T^{2} - 84 T^{3} + 238 T^{4} + 704 p T^{5} - 8591 T^{6} + 704 p^{3} T^{7} + 238 p^{4} T^{8} - 84 p^{6} T^{9} - 19 p^{8} T^{10} - 4 p^{10} T^{11} + p^{12} T^{12} \) |
| 7 | \( ( 1 + 11 T + 146 T^{2} + 1031 T^{3} + 146 p^{2} T^{4} + 11 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 11 | \( ( 1 + 18 T + 267 T^{2} + 3272 T^{3} + 267 p^{2} T^{4} + 18 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 13 | \( 1 + 3 T + 167 T^{2} + 492 T^{3} + 101 T^{4} + 65361 T^{5} - 4911130 T^{6} + 65361 p^{2} T^{7} + 101 p^{4} T^{8} + 492 p^{6} T^{9} + 167 p^{8} T^{10} + 3 p^{10} T^{11} + p^{12} T^{12} \) |
| 17 | \( 1 - 38 T + 541 T^{2} - 4834 T^{3} - 334 p T^{4} + 3349474 T^{5} - 90714787 T^{6} + 3349474 p^{2} T^{7} - 334 p^{5} T^{8} - 4834 p^{6} T^{9} + 541 p^{8} T^{10} - 38 p^{10} T^{11} + p^{12} T^{12} \) |
| 23 | \( 1 - 54 T + 765 T^{2} - 11066 T^{3} + 651546 T^{4} - 5242626 T^{5} - 206650371 T^{6} - 5242626 p^{2} T^{7} + 651546 p^{4} T^{8} - 11066 p^{6} T^{9} + 765 p^{8} T^{10} - 54 p^{10} T^{11} + p^{12} T^{12} \) |
| 29 | \( 1 + 102 T + 6203 T^{2} + 278970 T^{3} + 9713378 T^{4} + 292520358 T^{5} + 8574179483 T^{6} + 292520358 p^{2} T^{7} + 9713378 p^{4} T^{8} + 278970 p^{6} T^{9} + 6203 p^{8} T^{10} + 102 p^{10} T^{11} + p^{12} T^{12} \) |
| 31 | \( 1 - 3421 T^{2} + 5823134 T^{4} - 6536013745 T^{6} + 5823134 p^{4} T^{8} - 3421 p^{8} T^{10} + p^{12} T^{12} \) |
| 37 | \( 1 - 3501 T^{2} + 9576054 T^{4} - 14575249001 T^{6} + 9576054 p^{4} T^{8} - 3501 p^{8} T^{10} + p^{12} T^{12} \) |
| 41 | \( 1 - 96 T + 8867 T^{2} - 556320 T^{3} + 34233206 T^{4} - 1610757504 T^{5} + 73428522359 T^{6} - 1610757504 p^{2} T^{7} + 34233206 p^{4} T^{8} - 556320 p^{6} T^{9} + 8867 p^{8} T^{10} - 96 p^{10} T^{11} + p^{12} T^{12} \) |
| 43 | \( 1 - 107 T + 2315 T^{2} - 109092 T^{3} + 20090029 T^{4} - 716895265 T^{5} + 7701663934 T^{6} - 716895265 p^{2} T^{7} + 20090029 p^{4} T^{8} - 109092 p^{6} T^{9} + 2315 p^{8} T^{10} - 107 p^{10} T^{11} + p^{12} T^{12} \) |
| 47 | \( 1 + 50 T - 4651 T^{2} - 79434 T^{3} + 24415642 T^{4} + 226764442 T^{5} - 55863211187 T^{6} + 226764442 p^{2} T^{7} + 24415642 p^{4} T^{8} - 79434 p^{6} T^{9} - 4651 p^{8} T^{10} + 50 p^{10} T^{11} + p^{12} T^{12} \) |
| 53 | \( 1 + 90 T + 123 p T^{2} + 343710 T^{3} + 14892630 T^{4} + 694668222 T^{5} + 20439623059 T^{6} + 694668222 p^{2} T^{7} + 14892630 p^{4} T^{8} + 343710 p^{6} T^{9} + 123 p^{9} T^{10} + 90 p^{10} T^{11} + p^{12} T^{12} \) |
| 59 | \( 1 + 9395 T^{2} + 55562030 T^{4} + 82187460 T^{5} + 222918474911 T^{6} + 82187460 p^{2} T^{7} + 55562030 p^{4} T^{8} + 9395 p^{8} T^{10} + p^{12} T^{12} \) |
| 61 | \( 1 - 27 T - 8277 T^{2} + 88964 T^{3} + 43113009 T^{4} - 59789481 T^{5} - 181278951354 T^{6} - 59789481 p^{2} T^{7} + 43113009 p^{4} T^{8} + 88964 p^{6} T^{9} - 8277 p^{8} T^{10} - 27 p^{10} T^{11} + p^{12} T^{12} \) |
| 67 | \( 1 + 39 T + 7859 T^{2} + 286728 T^{3} + 23155985 T^{4} + 405814017 T^{5} + 60263661254 T^{6} + 405814017 p^{2} T^{7} + 23155985 p^{4} T^{8} + 286728 p^{6} T^{9} + 7859 p^{8} T^{10} + 39 p^{10} T^{11} + p^{12} T^{12} \) |
| 71 | \( 1 - 84 T + 10371 T^{2} - 673596 T^{3} + 28816422 T^{4} + 720991884 T^{5} + 18547655047 T^{6} + 720991884 p^{2} T^{7} + 28816422 p^{4} T^{8} - 673596 p^{6} T^{9} + 10371 p^{8} T^{10} - 84 p^{10} T^{11} + p^{12} T^{12} \) |
| 73 | \( 1 + 77 T - 1517 T^{2} + 250636 T^{3} + 3339001 T^{4} - 37295617 p T^{5} - 135125227066 T^{6} - 37295617 p^{3} T^{7} + 3339001 p^{4} T^{8} + 250636 p^{6} T^{9} - 1517 p^{8} T^{10} + 77 p^{10} T^{11} + p^{12} T^{12} \) |
| 79 | \( 1 - 9 T + 11711 T^{2} - 105156 T^{3} + 64875497 T^{4} - 4632616851 T^{5} + 425401555742 T^{6} - 4632616851 p^{2} T^{7} + 64875497 p^{4} T^{8} - 105156 p^{6} T^{9} + 11711 p^{8} T^{10} - 9 p^{10} T^{11} + p^{12} T^{12} \) |
| 83 | \( ( 1 + 174 T + 16527 T^{2} + 1223300 T^{3} + 16527 p^{2} T^{4} + 174 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 89 | \( 1 + 72 T + 12639 T^{2} + 785592 T^{3} + 28024818 T^{4} - 6608294460 T^{5} - 147146320001 T^{6} - 6608294460 p^{2} T^{7} + 28024818 p^{4} T^{8} + 785592 p^{6} T^{9} + 12639 p^{8} T^{10} + 72 p^{10} T^{11} + p^{12} T^{12} \) |
| 97 | \( 1 + 228 T + 49763 T^{2} + 7395180 T^{3} + 1074017366 T^{4} + 119443328292 T^{5} + 12875825313911 T^{6} + 119443328292 p^{2} T^{7} + 1074017366 p^{4} T^{8} + 7395180 p^{6} T^{9} + 49763 p^{8} T^{10} + 228 p^{10} T^{11} + p^{12} T^{12} \) |
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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
−8.410677293116563631383377751755, −8.176390519637391945755430688798, −7.917144563276463251773311126249, −7.65254742692637118163663675371, −7.22402978253919709120349939908, −7.19969846702714987664600363578, −7.11290330523685174443410523990, −6.53592337228709977535872297241, −6.52947701569784611419199066615, −6.07238600745051223249400648572, −5.91927260795359391394277318020, −5.76153876022377098222615039401, −5.57447388870786483395014204481, −5.33322032445728070169032650509, −5.23840760169883637494100121473, −4.86563072204190869620644816458, −4.78597676192681787709192215491, −4.13649794607816133337634663327, −4.07565409463688850198707509694, −3.43232400942395974462191543339, −3.10020345888458115510291274716, −2.74861974024507576704049969062, −2.74829494438792967821679412498, −1.27697268716449623729434958681, −0.10309517159171619023035595239,
0.10309517159171619023035595239, 1.27697268716449623729434958681, 2.74829494438792967821679412498, 2.74861974024507576704049969062, 3.10020345888458115510291274716, 3.43232400942395974462191543339, 4.07565409463688850198707509694, 4.13649794607816133337634663327, 4.78597676192681787709192215491, 4.86563072204190869620644816458, 5.23840760169883637494100121473, 5.33322032445728070169032650509, 5.57447388870786483395014204481, 5.76153876022377098222615039401, 5.91927260795359391394277318020, 6.07238600745051223249400648572, 6.52947701569784611419199066615, 6.53592337228709977535872297241, 7.11290330523685174443410523990, 7.19969846702714987664600363578, 7.22402978253919709120349939908, 7.65254742692637118163663675371, 7.917144563276463251773311126249, 8.176390519637391945755430688798, 8.410677293116563631383377751755
Plot not available for L-functions of degree greater than 10.
