| L(s) = 1 | + 3·3-s + 6·7-s + 3·9-s − 12·11-s − 3·13-s + 18·21-s + 3·25-s − 2·27-s − 12·29-s + 30·31-s − 36·33-s − 6·37-s − 9·39-s − 12·41-s − 9·43-s + 18·47-s + 3·49-s + 6·59-s + 3·61-s + 18·63-s + 3·67-s − 6·71-s + 9·73-s + 9·75-s − 72·77-s − 27·79-s − 9·81-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 2.26·7-s + 9-s − 3.61·11-s − 0.832·13-s + 3.92·21-s + 3/5·25-s − 0.384·27-s − 2.22·29-s + 5.38·31-s − 6.26·33-s − 0.986·37-s − 1.44·39-s − 1.87·41-s − 1.37·43-s + 2.62·47-s + 3/7·49-s + 0.781·59-s + 0.384·61-s + 2.26·63-s + 0.366·67-s − 0.712·71-s + 1.05·73-s + 1.03·75-s − 8.20·77-s − 3.03·79-s − 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4145307466\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4145307466\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 - T + T^{2} )^{3} \) |
| 19 | \( 1 + 9 T^{2} + 64 T^{3} + 9 p T^{4} + p^{3} T^{6} \) |
| good | 5 | \( 1 - 3 T^{2} + 16 T^{3} - 6 T^{4} - 24 T^{5} + 269 T^{6} - 24 p T^{7} - 6 p^{2} T^{8} + 16 p^{3} T^{9} - 3 p^{4} T^{10} + p^{6} T^{12} \) |
| 7 | \( ( 1 - 3 T + 12 T^{2} - 39 T^{3} + 12 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 11 | \( ( 1 + 6 T + 9 T^{2} - 4 T^{3} + 9 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( ( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} )^{3} \) |
| 17 | \( 1 - 3 T^{2} + 128 T^{3} - 42 T^{4} - 192 T^{5} + 13769 T^{6} - 192 p T^{7} - 42 p^{2} T^{8} + 128 p^{3} T^{9} - 3 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( 1 - 57 T^{2} + 16 T^{3} + 1938 T^{4} - 456 T^{5} - 50329 T^{6} - 456 p T^{7} + 1938 p^{2} T^{8} + 16 p^{3} T^{9} - 57 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( 1 + 12 T + 57 T^{2} + 36 T^{3} - 1002 T^{4} - 228 p T^{5} - 37811 T^{6} - 228 p^{2} T^{7} - 1002 p^{2} T^{8} + 36 p^{3} T^{9} + 57 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 31 | \( ( 1 - 15 T + 132 T^{2} - 803 T^{3} + 132 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( ( 1 + 3 T + 66 T^{2} + 3 p T^{3} + 66 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 41 | \( 1 + 12 T + 21 T^{2} - 108 T^{3} + 582 T^{4} - 84 p T^{5} - 97247 T^{6} - 84 p^{2} T^{7} + 582 p^{2} T^{8} - 108 p^{3} T^{9} + 21 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 43 | \( 1 + 9 T - 63 T^{2} - 218 T^{3} + 7731 T^{4} + 14193 T^{5} - 309354 T^{6} + 14193 p T^{7} + 7731 p^{2} T^{8} - 218 p^{3} T^{9} - 63 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 47 | \( ( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{3} \) |
| 53 | \( 1 - 75 T^{2} + 592 T^{3} + 1650 T^{4} - 22200 T^{5} + 87245 T^{6} - 22200 p T^{7} + 1650 p^{2} T^{8} + 592 p^{3} T^{9} - 75 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( 1 - 6 T - 45 T^{2} + 82 T^{3} + 78 T^{4} + 13458 T^{5} - 21001 T^{6} + 13458 p T^{7} + 78 p^{2} T^{8} + 82 p^{3} T^{9} - 45 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 61 | \( 1 - 3 T - 129 T^{2} + 352 T^{3} + 9477 T^{4} - 13509 T^{5} - 594522 T^{6} - 13509 p T^{7} + 9477 p^{2} T^{8} + 352 p^{3} T^{9} - 129 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 67 | \( 1 - 3 T - 159 T^{2} + 374 T^{3} + 15651 T^{4} - 19683 T^{5} - 1137162 T^{6} - 19683 p T^{7} + 15651 p^{2} T^{8} + 374 p^{3} T^{9} - 159 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 71 | \( 1 + 6 T - 45 T^{2} + 494 T^{3} + 834 T^{4} - 39090 T^{5} + 68531 T^{6} - 39090 p T^{7} + 834 p^{2} T^{8} + 494 p^{3} T^{9} - 45 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 73 | \( 1 - 9 T - 117 T^{2} + 484 T^{3} + 14553 T^{4} - 12123 T^{5} - 1231818 T^{6} - 12123 p T^{7} + 14553 p^{2} T^{8} + 484 p^{3} T^{9} - 117 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 79 | \( 1 + 27 T + 333 T^{2} + 2806 T^{3} + 19071 T^{4} + 40599 T^{5} - 404826 T^{6} + 40599 p T^{7} + 19071 p^{2} T^{8} + 2806 p^{3} T^{9} + 333 p^{4} T^{10} + 27 p^{5} T^{11} + p^{6} T^{12} \) |
| 83 | \( ( 1 + 12 T + 153 T^{2} + 2056 T^{3} + 153 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 89 | \( 1 + 18 T - 15 T^{2} - 162 T^{3} + 28374 T^{4} + 118170 T^{5} - 1575011 T^{6} + 118170 p T^{7} + 28374 p^{2} T^{8} - 162 p^{3} T^{9} - 15 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \) |
| 97 | \( 1 + 6 T - 123 T^{2} + 338 T^{3} + 8010 T^{4} - 79650 T^{5} - 870771 T^{6} - 79650 p T^{7} + 8010 p^{2} T^{8} + 338 p^{3} T^{9} - 123 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
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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
−5.87152159929193811168158049945, −5.65155214137333701645396386067, −5.36732885438047691826820640368, −5.35743662369542022094202757015, −5.27586477574640722538741545288, −5.23726987411277387711568767612, −4.80470271464506886205789532181, −4.72647331185169874286615277883, −4.67809435462431667509939511686, −4.31612378072655655151866658651, −4.20356081832464200812901194295, −3.94391880622174462609989476933, −3.82655522276379584557257224375, −3.36406422696934501525334580603, −3.13279296713759848666879749541, −2.89212986225982135731192086143, −2.87354649692896317311034782538, −2.61221931917186251718164038700, −2.49383525775058505295071150770, −2.19654629944842660737891754276, −2.19520167911988459636789314366, −1.54824863182356726832785591831, −1.36449277396127253475454976398, −1.23892178352253258141244413606, −0.11406657696082868091680608505,
0.11406657696082868091680608505, 1.23892178352253258141244413606, 1.36449277396127253475454976398, 1.54824863182356726832785591831, 2.19520167911988459636789314366, 2.19654629944842660737891754276, 2.49383525775058505295071150770, 2.61221931917186251718164038700, 2.87354649692896317311034782538, 2.89212986225982135731192086143, 3.13279296713759848666879749541, 3.36406422696934501525334580603, 3.82655522276379584557257224375, 3.94391880622174462609989476933, 4.20356081832464200812901194295, 4.31612378072655655151866658651, 4.67809435462431667509939511686, 4.72647331185169874286615277883, 4.80470271464506886205789532181, 5.23726987411277387711568767612, 5.27586477574640722538741545288, 5.35743662369542022094202757015, 5.36732885438047691826820640368, 5.65155214137333701645396386067, 5.87152159929193811168158049945
Plot not available for L-functions of degree greater than 10.
