| L(s) = 1 | − 3·2-s + 3·3-s + 3·4-s − 9·6-s + 2·8-s + 3·9-s + 3·11-s + 9·12-s − 9·16-s − 3·17-s − 9·18-s + 9·19-s − 9·22-s − 3·23-s + 6·24-s − 2·27-s + 18·29-s + 6·31-s + 9·32-s + 9·33-s + 9·34-s + 9·36-s − 21·37-s − 27·38-s + 24·41-s + 6·43-s + 9·44-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 1.73·3-s + 3/2·4-s − 3.67·6-s + 0.707·8-s + 9-s + 0.904·11-s + 2.59·12-s − 9/4·16-s − 0.727·17-s − 2.12·18-s + 2.06·19-s − 1.91·22-s − 0.625·23-s + 1.22·24-s − 0.384·27-s + 3.34·29-s + 1.07·31-s + 1.59·32-s + 1.56·33-s + 1.54·34-s + 3/2·36-s − 3.45·37-s − 4.37·38-s + 3.74·41-s + 0.914·43-s + 1.35·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 7^{6} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 7^{6} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.262733685\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.262733685\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T + T^{2} )^{3} \) |
| 3 | \( ( 1 - T + T^{2} )^{3} \) |
| 7 | \( 1 + 6 T^{2} - 20 T^{3} + 6 p T^{4} + p^{3} T^{6} \) |
| 11 | \( ( 1 - T + T^{2} )^{3} \) |
| good | 5 | \( ( 1 - 4 p T^{3} + p^{3} T^{6} )^{2} \) |
| 13 | \( ( 1 + p T^{2} )^{6} \) |
| 17 | \( 1 + 3 T + 15 T^{2} + 56 T^{3} + 249 T^{4} + 1701 T^{5} + 7982 T^{6} + 1701 p T^{7} + 249 p^{2} T^{8} + 56 p^{3} T^{9} + 15 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 19 | \( 1 - 9 T + 12 T^{2} + 7 T^{3} + 624 T^{4} - 1317 T^{5} - 7386 T^{6} - 1317 p T^{7} + 624 p^{2} T^{8} + 7 p^{3} T^{9} + 12 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 23 | \( 1 + 3 T - 33 T^{2} + 28 T^{3} + 651 T^{4} - 2151 T^{5} - 18598 T^{6} - 2151 p T^{7} + 651 p^{2} T^{8} + 28 p^{3} T^{9} - 33 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 29 | \( ( 1 - 9 T + 39 T^{2} - 6 p T^{3} + 39 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( 1 - 6 T - 9 T^{2} + 90 T^{3} - 450 T^{4} + 2874 T^{5} - 5389 T^{6} + 2874 p T^{7} - 450 p^{2} T^{8} + 90 p^{3} T^{9} - 9 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 37 | \( 1 + 21 T + 198 T^{2} + 1539 T^{3} + 12636 T^{4} + 84153 T^{5} + 488288 T^{6} + 84153 p T^{7} + 12636 p^{2} T^{8} + 1539 p^{3} T^{9} + 198 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \) |
| 41 | \( ( 1 - 12 T + 96 T^{2} - 678 T^{3} + 96 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( ( 1 - 3 T - 3 T^{2} + 146 T^{3} - 3 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 47 | \( 1 + 3 T - 105 T^{2} - 204 T^{3} + 6819 T^{4} + 5601 T^{5} - 348518 T^{6} + 5601 p T^{7} + 6819 p^{2} T^{8} - 204 p^{3} T^{9} - 105 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 53 | \( ( 1 + 8 T + 11 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{3} \) |
| 59 | \( 1 - 9 T - 48 T^{2} + 867 T^{3} - 96 T^{4} - 25137 T^{5} + 175174 T^{6} - 25137 p T^{7} - 96 p^{2} T^{8} + 867 p^{3} T^{9} - 48 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 61 | \( 1 - 6 T - 84 T^{2} + 940 T^{3} + 2100 T^{4} - 34086 T^{5} + 130506 T^{6} - 34086 p T^{7} + 2100 p^{2} T^{8} + 940 p^{3} T^{9} - 84 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 67 | \( 1 + 6 T - 102 T^{2} - 356 T^{3} + 6246 T^{4} - 2322 T^{5} - 485562 T^{6} - 2322 p T^{7} + 6246 p^{2} T^{8} - 356 p^{3} T^{9} - 102 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 71 | \( ( 1 - 9 T + 105 T^{2} - 1170 T^{3} + 105 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 99 T^{2} - 960 T^{3} + 2574 T^{4} + 47520 T^{5} + 292961 T^{6} + 47520 p T^{7} + 2574 p^{2} T^{8} - 960 p^{3} T^{9} - 99 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( 1 + 12 T - 126 T^{2} - 520 T^{3} + 29010 T^{4} + 70572 T^{5} - 2137326 T^{6} + 70572 p T^{7} + 29010 p^{2} T^{8} - 520 p^{3} T^{9} - 126 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 83 | \( ( 1 - 18 T + 282 T^{2} - 2824 T^{3} + 282 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 89 | \( 1 - 12 T - 111 T^{2} + 1180 T^{3} + 16890 T^{4} - 90972 T^{5} - 1023511 T^{6} - 90972 p T^{7} + 16890 p^{2} T^{8} + 1180 p^{3} T^{9} - 111 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 97 | \( ( 1 - 7 T + p T^{2} )^{6} \) |
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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
−6.04882091979215955060644518988, −5.80841543241917664499988665434, −5.70074183232271459602453796950, −5.68696509750204597620133698352, −4.86822988435747372158991715480, −4.85980901983153694818112454399, −4.83088927317290942119420186579, −4.78985201766872580628018039073, −4.77125096178599153164923953251, −4.42638206651835522727959198060, −3.81247630022705073193174997289, −3.79624440550515185179756374386, −3.65099212679395032654023090375, −3.46121201806650984943637691595, −3.44190066245201312876857889554, −2.88994144937451180846727587660, −2.78812124612115249260725329055, −2.58389578842827060809559373832, −2.43674965623233933892798906181, −1.82406048634981662875060139327, −1.80893258573791624009973629299, −1.76492509313513827125164267848, −0.900292199053185523838336862067, −0.879751540973423425940944629942, −0.67753141026476462879431093064,
0.67753141026476462879431093064, 0.879751540973423425940944629942, 0.900292199053185523838336862067, 1.76492509313513827125164267848, 1.80893258573791624009973629299, 1.82406048634981662875060139327, 2.43674965623233933892798906181, 2.58389578842827060809559373832, 2.78812124612115249260725329055, 2.88994144937451180846727587660, 3.44190066245201312876857889554, 3.46121201806650984943637691595, 3.65099212679395032654023090375, 3.79624440550515185179756374386, 3.81247630022705073193174997289, 4.42638206651835522727959198060, 4.77125096178599153164923953251, 4.78985201766872580628018039073, 4.83088927317290942119420186579, 4.85980901983153694818112454399, 4.86822988435747372158991715480, 5.68696509750204597620133698352, 5.70074183232271459602453796950, 5.80841543241917664499988665434, 6.04882091979215955060644518988
Plot not available for L-functions of degree greater than 10.
