| L(s) = 1 | − 3-s + 6·5-s − 5·7-s + 5·9-s + 3·11-s + 15·13-s − 6·15-s + 5·17-s − 4·19-s + 5·21-s + 23-s + 9·25-s − 4·27-s − 4·29-s + 14·31-s − 3·33-s − 30·35-s + 8·37-s − 15·39-s + 5·41-s − 8·43-s + 30·45-s + 12·47-s + 17·49-s − 5·51-s + 22·53-s + 18·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 2.68·5-s − 1.88·7-s + 5/3·9-s + 0.904·11-s + 4.16·13-s − 1.54·15-s + 1.21·17-s − 0.917·19-s + 1.09·21-s + 0.208·23-s + 9/5·25-s − 0.769·27-s − 0.742·29-s + 2.51·31-s − 0.522·33-s − 5.07·35-s + 1.31·37-s − 2.40·39-s + 0.780·41-s − 1.21·43-s + 4.47·45-s + 1.75·47-s + 17/7·49-s − 0.700·51-s + 3.02·53-s + 2.42·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 11^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 11^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.420447520\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.420447520\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( ( 1 - T + T^{2} )^{3} \) |
| 13 | \( ( 1 - 5 T + p T^{2} )^{3} \) |
| good | 3 | \( 1 + T - 4 T^{2} - 5 T^{3} + 5 T^{4} + 4 T^{5} - 5 T^{6} + 4 p T^{7} + 5 p^{2} T^{8} - 5 p^{3} T^{9} - 4 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \) |
| 5 | \( ( 1 - 3 T + 9 T^{2} - 21 T^{3} + 9 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 7 | \( 1 + 5 T + 8 T^{2} + 3 T^{3} - 5 p T^{4} - 248 T^{5} - 881 T^{6} - 248 p T^{7} - 5 p^{3} T^{8} + 3 p^{3} T^{9} + 8 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \) |
| 17 | \( 1 - 5 T - 8 T^{2} + 13 T^{3} + 83 T^{4} + 1228 T^{5} - 7991 T^{6} + 1228 p T^{7} + 83 p^{2} T^{8} + 13 p^{3} T^{9} - 8 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \) |
| 19 | \( 1 + 4 T - 42 T^{2} - 66 T^{3} + 1666 T^{4} + 1270 T^{5} - 33773 T^{6} + 1270 p T^{7} + 1666 p^{2} T^{8} - 66 p^{3} T^{9} - 42 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \) |
| 23 | \( 1 - T - 14 T^{2} + 263 T^{3} - 259 T^{4} - 1924 T^{5} + 37951 T^{6} - 1924 p T^{7} - 259 p^{2} T^{8} + 263 p^{3} T^{9} - 14 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \) |
| 29 | \( 1 + 4 T - 26 T^{2} + 2 p T^{3} + 326 T^{4} - 4070 T^{5} - 12809 T^{6} - 4070 p T^{7} + 326 p^{2} T^{8} + 2 p^{4} T^{9} - 26 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \) |
| 31 | \( ( 1 - 7 T + 40 T^{2} - 311 T^{3} + 40 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 8 T - 12 T^{2} + 630 T^{3} - 1988 T^{4} - 13004 T^{5} + 183139 T^{6} - 13004 p T^{7} - 1988 p^{2} T^{8} + 630 p^{3} T^{9} - 12 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \) |
| 41 | \( 1 - 5 T - 80 T^{2} + 133 T^{3} + 5075 T^{4} + 100 T^{5} - 247967 T^{6} + 100 p T^{7} + 5075 p^{2} T^{8} + 133 p^{3} T^{9} - 80 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \) |
| 43 | \( 1 + 8 T + 68 T^{2} + 846 T^{3} + 3700 T^{4} + 19300 T^{5} + 232711 T^{6} + 19300 p T^{7} + 3700 p^{2} T^{8} + 846 p^{3} T^{9} + 68 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \) |
| 47 | \( ( 1 - 6 T + 126 T^{2} - 465 T^{3} + 126 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 53 | \( ( 1 - 11 T + 3 p T^{2} - 1163 T^{3} + 3 p^{2} T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 59 | \( 1 + 22 T + 196 T^{2} + 1318 T^{3} + 10754 T^{4} + 51526 T^{5} + 127339 T^{6} + 51526 p T^{7} + 10754 p^{2} T^{8} + 1318 p^{3} T^{9} + 196 p^{4} T^{10} + 22 p^{5} T^{11} + p^{6} T^{12} \) |
| 61 | \( 1 + 6 T - 90 T^{2} - 826 T^{3} + 3444 T^{4} + 29760 T^{5} - 22749 T^{6} + 29760 p T^{7} + 3444 p^{2} T^{8} - 826 p^{3} T^{9} - 90 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 67 | \( 1 + 7 T - 84 T^{2} - 819 T^{3} + 2653 T^{4} + 23044 T^{5} - 37973 T^{6} + 23044 p T^{7} + 2653 p^{2} T^{8} - 819 p^{3} T^{9} - 84 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \) |
| 71 | \( 1 - 11 T - 26 T^{2} + 349 T^{3} + 359 T^{4} + 29662 T^{5} - 462089 T^{6} + 29662 p T^{7} + 359 p^{2} T^{8} + 349 p^{3} T^{9} - 26 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \) |
| 73 | \( ( 1 - 2 T + 119 T^{2} + 100 T^{3} + 119 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 79 | \( ( 1 + 20 T + 145 T^{2} + 688 T^{3} + 145 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( ( 1 - 2 T + 210 T^{2} - 401 T^{3} + 210 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 89 | \( 1 + 27 T + 300 T^{2} + 2025 T^{3} + 16827 T^{4} + 202284 T^{5} + 2094721 T^{6} + 202284 p T^{7} + 16827 p^{2} T^{8} + 2025 p^{3} T^{9} + 300 p^{4} T^{10} + 27 p^{5} T^{11} + p^{6} T^{12} \) |
| 97 | \( 1 + 7 T - 189 T^{2} - 804 T^{3} + 24913 T^{4} + 36589 T^{5} - 2604938 T^{6} + 36589 p T^{7} + 24913 p^{2} T^{8} - 804 p^{3} T^{9} - 189 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \) |
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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
−6.01207414443416946951908129223, −5.68588568596341240854600143896, −5.60886230080885425077968322097, −5.39642297922255685461507255985, −5.21138192646605211975012087003, −4.96472659384464100421725485934, −4.85634244555337405643000134398, −4.16591911226042070490529088578, −4.16119764073536957347489721034, −4.16030063555240686901945192132, −3.98739227565317839536299742195, −3.97739492828267563469898087458, −3.78344245502699430468359565177, −3.34601554392530329748395855602, −3.16776128107826727126893445832, −2.79369619696306451321851505813, −2.65022169667188793686340609877, −2.62995502362101678239244085635, −2.44384408799601666545438837678, −1.72041292009131241352783617731, −1.51671407712518381077924802123, −1.48273980454248708120301866805, −1.23175758434516573003712055196, −1.20806776865288831794180790294, −0.53697340599067567078445436559,
0.53697340599067567078445436559, 1.20806776865288831794180790294, 1.23175758434516573003712055196, 1.48273980454248708120301866805, 1.51671407712518381077924802123, 1.72041292009131241352783617731, 2.44384408799601666545438837678, 2.62995502362101678239244085635, 2.65022169667188793686340609877, 2.79369619696306451321851505813, 3.16776128107826727126893445832, 3.34601554392530329748395855602, 3.78344245502699430468359565177, 3.97739492828267563469898087458, 3.98739227565317839536299742195, 4.16030063555240686901945192132, 4.16119764073536957347489721034, 4.16591911226042070490529088578, 4.85634244555337405643000134398, 4.96472659384464100421725485934, 5.21138192646605211975012087003, 5.39642297922255685461507255985, 5.60886230080885425077968322097, 5.68588568596341240854600143896, 6.01207414443416946951908129223
Plot not available for L-functions of degree greater than 10.
