| L(s) = 1 | − 3·3-s − 6·5-s − 2·7-s + 3·9-s + 2·11-s + 5·13-s + 18·15-s + 3·17-s + 5·19-s + 6·21-s + 14·23-s + 23·25-s + 2·27-s + 6·29-s + 20·31-s − 6·33-s + 12·35-s + 3·37-s − 15·39-s + 6·41-s − 22·43-s − 18·45-s + 8·47-s + 10·49-s − 9·51-s + 10·53-s − 12·55-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 2.68·5-s − 0.755·7-s + 9-s + 0.603·11-s + 1.38·13-s + 4.64·15-s + 0.727·17-s + 1.14·19-s + 1.30·21-s + 2.91·23-s + 23/5·25-s + 0.384·27-s + 1.11·29-s + 3.59·31-s − 1.04·33-s + 2.02·35-s + 0.493·37-s − 2.40·39-s + 0.937·41-s − 3.35·43-s − 2.68·45-s + 1.16·47-s + 10/7·49-s − 1.26·51-s + 1.37·53-s − 1.61·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{6} \cdot 37^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{6} \cdot 37^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.426862983\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.426862983\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + T + T^{2} )^{3} \) |
| 37 | \( 1 - 3 T - 30 T^{2} + 137 T^{3} - 30 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| good | 5 | \( 1 + 6 T + 13 T^{2} + 16 T^{3} + 48 T^{4} + 173 T^{5} + 426 T^{6} + 173 p T^{7} + 48 p^{2} T^{8} + 16 p^{3} T^{9} + 13 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 7 | \( 1 + 2 T - 6 T^{2} - 44 T^{3} - 34 T^{4} + 138 T^{5} + 814 T^{6} + 138 p T^{7} - 34 p^{2} T^{8} - 44 p^{3} T^{9} - 6 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \) |
| 11 | \( ( 1 - T + 14 T^{2} + 10 T^{3} + 14 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 - 5 T - 3 T^{2} + 116 T^{3} - 220 T^{4} - 789 T^{5} + 6361 T^{6} - 789 p T^{7} - 220 p^{2} T^{8} + 116 p^{3} T^{9} - 3 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \) |
| 17 | \( ( 1 - T - 16 T^{2} - p T^{3} + p^{2} T^{4} )^{3} \) |
| 19 | \( 1 - 5 T - 9 T^{2} + 98 T^{3} - 160 T^{4} + 99 T^{5} + 685 T^{6} + 99 p T^{7} - 160 p^{2} T^{8} + 98 p^{3} T^{9} - 9 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \) |
| 23 | \( ( 1 - 7 T + 80 T^{2} - 326 T^{3} + 80 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 29 | \( ( 1 - 3 T + 74 T^{2} - 167 T^{3} + 74 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( ( 1 - 10 T + 105 T^{2} - 556 T^{3} + 105 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 41 | \( 1 - 6 T + 39 T^{2} - 772 T^{3} + 2502 T^{4} - 12747 T^{5} + 232982 T^{6} - 12747 p T^{7} + 2502 p^{2} T^{8} - 772 p^{3} T^{9} + 39 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 43 | \( ( 1 + 11 T + 78 T^{2} + 362 T^{3} + 78 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 47 | \( ( 1 - 4 T + 140 T^{2} - 368 T^{3} + 140 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 53 | \( 1 - 10 T - 86 T^{2} + 288 T^{3} + 13998 T^{4} - 30694 T^{5} - 663258 T^{6} - 30694 p T^{7} + 13998 p^{2} T^{8} + 288 p^{3} T^{9} - 86 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \) |
| 59 | \( 1 - 5 T - p T^{2} + 168 T^{3} + 630 T^{4} + 7459 T^{5} - 6135 T^{6} + 7459 p T^{7} + 630 p^{2} T^{8} + 168 p^{3} T^{9} - p^{5} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \) |
| 61 | \( 1 - 3 T - 120 T^{2} + 129 T^{3} + 7998 T^{4} + 2769 T^{5} - 528064 T^{6} + 2769 p T^{7} + 7998 p^{2} T^{8} + 129 p^{3} T^{9} - 120 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 67 | \( 1 - 13 T - 61 T^{2} + 310 T^{3} + 16040 T^{4} - 37197 T^{5} - 978915 T^{6} - 37197 p T^{7} + 16040 p^{2} T^{8} + 310 p^{3} T^{9} - 61 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} \) |
| 71 | \( 1 - 41 T^{2} + 1696 T^{3} - 1230 T^{4} - 34768 T^{5} + 1315575 T^{6} - 34768 p T^{7} - 1230 p^{2} T^{8} + 1696 p^{3} T^{9} - 41 p^{4} T^{10} + p^{6} T^{12} \) |
| 73 | \( ( 1 - 10 T + 167 T^{2} - 1452 T^{3} + 167 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 79 | \( 1 + 19 T + 61 T^{2} + 88 T^{3} + 12986 T^{4} + 59779 T^{5} - 503287 T^{6} + 59779 p T^{7} + 12986 p^{2} T^{8} + 88 p^{3} T^{9} + 61 p^{4} T^{10} + 19 p^{5} T^{11} + p^{6} T^{12} \) |
| 83 | \( 1 + 4 T - 153 T^{2} - 908 T^{3} + 11366 T^{4} + 49188 T^{5} - 709949 T^{6} + 49188 p T^{7} + 11366 p^{2} T^{8} - 908 p^{3} T^{9} - 153 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \) |
| 89 | \( 1 + 13 T - 32 T^{2} - 87 T^{3} + 4962 T^{4} - 90791 T^{5} - 1701900 T^{6} - 90791 p T^{7} + 4962 p^{2} T^{8} - 87 p^{3} T^{9} - 32 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12} \) |
| 97 | \( ( 1 - 34 T + 657 T^{2} - 7807 T^{3} + 657 p T^{4} - 34 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
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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
−5.44752175149271020257087566469, −5.07919715553025768287825794319, −4.90592895144130401539083726473, −4.90490464559849943491999678164, −4.80384524174888323713087671121, −4.73806684653641264364659654318, −4.35102614845750616385698217285, −4.13510600588180049964624044601, −4.05309762642077659148834512358, −3.95326817319987877986038630577, −3.60361822077170046845808179227, −3.51884986868490161928803798729, −3.32751819914484237480012741642, −3.29419684563654373188724205973, −2.92355763999627753147206453747, −2.87402625398870310715197236017, −2.49704648241644259740919542196, −2.48343589478493301465091938661, −2.28662652110126577990155150824, −1.25799541532834659750056024548, −1.25656852633820591506467553250, −1.22772634270977437711403151302, −0.892265099160117569649908696293, −0.71218923006885170924793780413, −0.36844509248436165341005512074,
0.36844509248436165341005512074, 0.71218923006885170924793780413, 0.892265099160117569649908696293, 1.22772634270977437711403151302, 1.25656852633820591506467553250, 1.25799541532834659750056024548, 2.28662652110126577990155150824, 2.48343589478493301465091938661, 2.49704648241644259740919542196, 2.87402625398870310715197236017, 2.92355763999627753147206453747, 3.29419684563654373188724205973, 3.32751819914484237480012741642, 3.51884986868490161928803798729, 3.60361822077170046845808179227, 3.95326817319987877986038630577, 4.05309762642077659148834512358, 4.13510600588180049964624044601, 4.35102614845750616385698217285, 4.73806684653641264364659654318, 4.80384524174888323713087671121, 4.90490464559849943491999678164, 4.90592895144130401539083726473, 5.07919715553025768287825794319, 5.44752175149271020257087566469
Plot not available for L-functions of degree greater than 10.
