| L(s) = 1 | − 15·5-s + 3·7-s + 24·11-s − 3·13-s − 60·17-s − 54·19-s − 99·23-s + 75·25-s + 54·29-s − 72·31-s − 45·35-s − 36·37-s − 411·41-s − 444·43-s + 75·47-s + 417·49-s + 342·53-s − 360·55-s − 297·59-s − 684·61-s + 45·65-s − 12·67-s + 1.28e3·71-s − 132·73-s + 72·77-s − 1.12e3·79-s − 90·83-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 0.161·7-s + 0.657·11-s − 0.0640·13-s − 0.856·17-s − 0.652·19-s − 0.897·23-s + 3/5·25-s + 0.345·29-s − 0.417·31-s − 0.217·35-s − 0.159·37-s − 1.56·41-s − 1.57·43-s + 0.232·47-s + 1.21·49-s + 0.886·53-s − 0.882·55-s − 0.655·59-s − 1.43·61-s + 0.0858·65-s − 0.0218·67-s + 2.14·71-s − 0.211·73-s + 0.106·77-s − 1.59·79-s − 0.119·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{24} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{24} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.01613208443\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01613208443\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( ( 1 + p T + p^{2} T^{2} )^{3} \) |
| good | 7 | \( 1 - 3 T - 408 T^{2} + 4337 T^{3} + 22476 T^{4} - 659859 T^{5} + 27810882 T^{6} - 659859 p^{3} T^{7} + 22476 p^{6} T^{8} + 4337 p^{9} T^{9} - 408 p^{12} T^{10} - 3 p^{15} T^{11} + p^{18} T^{12} \) |
| 11 | \( 1 - 24 T - 1992 T^{2} + 31728 T^{3} + 186936 p T^{4} + 1672680 T^{5} - 2902945898 T^{6} + 1672680 p^{3} T^{7} + 186936 p^{7} T^{8} + 31728 p^{9} T^{9} - 1992 p^{12} T^{10} - 24 p^{15} T^{11} + p^{18} T^{12} \) |
| 13 | \( 1 + 3 T - 54 p T^{2} - 9991 T^{3} - 1074732 T^{4} + 9418107 T^{5} + 20091675672 T^{6} + 9418107 p^{3} T^{7} - 1074732 p^{6} T^{8} - 9991 p^{9} T^{9} - 54 p^{13} T^{10} + 3 p^{15} T^{11} + p^{18} T^{12} \) |
| 17 | \( ( 1 + 30 T + 8571 T^{2} + 192828 T^{3} + 8571 p^{3} T^{4} + 30 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 19 | \( ( 1 + 27 T + 10608 T^{2} + 34717 p T^{3} + 10608 p^{3} T^{4} + 27 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 23 | \( 1 + 99 T - 29352 T^{2} - 970569 T^{3} + 816670416 T^{4} + 15215127687 T^{5} - 10627423224302 T^{6} + 15215127687 p^{3} T^{7} + 816670416 p^{6} T^{8} - 970569 p^{9} T^{9} - 29352 p^{12} T^{10} + 99 p^{15} T^{11} + p^{18} T^{12} \) |
| 29 | \( 1 - 54 T - 70608 T^{2} + 1289952 T^{3} + 3470575044 T^{4} - 31834598022 T^{5} - 97644194286986 T^{6} - 31834598022 p^{3} T^{7} + 3470575044 p^{6} T^{8} + 1289952 p^{9} T^{9} - 70608 p^{12} T^{10} - 54 p^{15} T^{11} + p^{18} T^{12} \) |
| 31 | \( 1 + 72 T - 68280 T^{2} - 1517320 T^{3} + 2963351064 T^{4} - 1228509528 T^{5} - 100919502631602 T^{6} - 1228509528 p^{3} T^{7} + 2963351064 p^{6} T^{8} - 1517320 p^{9} T^{9} - 68280 p^{12} T^{10} + 72 p^{15} T^{11} + p^{18} T^{12} \) |
| 37 | \( ( 1 + 18 T + 104499 T^{2} - 1140212 T^{3} + 104499 p^{3} T^{4} + 18 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 41 | \( 1 + 411 T - 73977 T^{2} - 11118432 T^{3} + 19458541041 T^{4} + 1666084429365 T^{5} - 1136480341638098 T^{6} + 1666084429365 p^{3} T^{7} + 19458541041 p^{6} T^{8} - 11118432 p^{9} T^{9} - 73977 p^{12} T^{10} + 411 p^{15} T^{11} + p^{18} T^{12} \) |
| 43 | \( 1 + 444 T - 34605 T^{2} - 37783108 T^{3} + 4048762530 T^{4} + 2745468491196 T^{5} + 436623851528091 T^{6} + 2745468491196 p^{3} T^{7} + 4048762530 p^{6} T^{8} - 37783108 p^{9} T^{9} - 34605 p^{12} T^{10} + 444 p^{15} T^{11} + p^{18} T^{12} \) |
| 47 | \( 1 - 75 T - 174684 T^{2} + 24209925 T^{3} + 12376122924 T^{4} - 1402216717575 T^{5} - 841849814761454 T^{6} - 1402216717575 p^{3} T^{7} + 12376122924 p^{6} T^{8} + 24209925 p^{9} T^{9} - 174684 p^{12} T^{10} - 75 p^{15} T^{11} + p^{18} T^{12} \) |
| 53 | \( ( 1 - 171 T + 455763 T^{2} - 51067422 T^{3} + 455763 p^{3} T^{4} - 171 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 59 | \( 1 + 297 T - 29919 T^{2} + 188413830 T^{3} + 12462691251 T^{4} - 10939693886847 T^{5} + 16616892260928022 T^{6} - 10939693886847 p^{3} T^{7} + 12462691251 p^{6} T^{8} + 188413830 p^{9} T^{9} - 29919 p^{12} T^{10} + 297 p^{15} T^{11} + p^{18} T^{12} \) |
| 61 | \( 1 + 684 T - 63027 T^{2} - 265320364 T^{3} - 53591557446 T^{4} + 610010798364 p T^{5} + 34048157230279581 T^{6} + 610010798364 p^{4} T^{7} - 53591557446 p^{6} T^{8} - 265320364 p^{9} T^{9} - 63027 p^{12} T^{10} + 684 p^{15} T^{11} + p^{18} T^{12} \) |
| 67 | \( 1 + 12 T - 698493 T^{2} - 53055508 T^{3} + 277600862826 T^{4} + 17504353163196 T^{5} - 91456777656400413 T^{6} + 17504353163196 p^{3} T^{7} + 277600862826 p^{6} T^{8} - 53055508 p^{9} T^{9} - 698493 p^{12} T^{10} + 12 p^{15} T^{11} + p^{18} T^{12} \) |
| 71 | \( ( 1 - 642 T + 197304 T^{2} + 87694746 T^{3} + 197304 p^{3} T^{4} - 642 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 73 | \( ( 1 + 66 T + 335715 T^{2} - 146251676 T^{3} + 335715 p^{3} T^{4} + 66 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 79 | \( 1 + 1122 T - 428073 T^{2} - 266737918 T^{3} + 802606767534 T^{4} + 227824373924802 T^{5} - 303553868663922141 T^{6} + 227824373924802 p^{3} T^{7} + 802606767534 p^{6} T^{8} - 266737918 p^{9} T^{9} - 428073 p^{12} T^{10} + 1122 p^{15} T^{11} + p^{18} T^{12} \) |
| 83 | \( 1 + 90 T - 1227801 T^{2} - 6466230 T^{3} + 815482777314 T^{4} - 301749773130 p T^{5} - 72388057449929 p^{2} T^{6} - 301749773130 p^{4} T^{7} + 815482777314 p^{6} T^{8} - 6466230 p^{9} T^{9} - 1227801 p^{12} T^{10} + 90 p^{15} T^{11} + p^{18} T^{12} \) |
| 89 | \( ( 1 - 756 T + 1118256 T^{2} - 284981598 T^{3} + 1118256 p^{3} T^{4} - 756 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 97 | \( 1 + 1536 T - 673023 T^{2} - 878940928 T^{3} + 2175347162658 T^{4} + 1171979614210944 T^{5} - 1173318191949724767 T^{6} + 1171979614210944 p^{3} T^{7} + 2175347162658 p^{6} T^{8} - 878940928 p^{9} T^{9} - 673023 p^{12} T^{10} + 1536 p^{15} T^{11} + p^{18} T^{12} \) |
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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
−4.45529678091106265882539483537, −4.35368256992059494133438665970, −4.31663946479509068026272547726, −4.21318395765081452597108142142, −3.71831926846712717981291729930, −3.66499593258028599044560419137, −3.56206085598588991731149537214, −3.52642814685450245174741473598, −3.39858832375687694422553744538, −3.30550753788332429831220007853, −2.82908572314313599213347925021, −2.68198264763332704239522575728, −2.42541926091310421797203343965, −2.34326440294617030359863329646, −2.26081237301735907043729459328, −2.19038118508490188687029649295, −1.84901735698239783177572304228, −1.46251935825834307852317716759, −1.24769914146808889867007860958, −1.20316334822528967394845319659, −1.07127249158505783498009172049, −1.07120086725833759604805790405, −0.25984707521403966053905458945, −0.11321405753354811904105281244, −0.05518089452995391561746595447,
0.05518089452995391561746595447, 0.11321405753354811904105281244, 0.25984707521403966053905458945, 1.07120086725833759604805790405, 1.07127249158505783498009172049, 1.20316334822528967394845319659, 1.24769914146808889867007860958, 1.46251935825834307852317716759, 1.84901735698239783177572304228, 2.19038118508490188687029649295, 2.26081237301735907043729459328, 2.34326440294617030359863329646, 2.42541926091310421797203343965, 2.68198264763332704239522575728, 2.82908572314313599213347925021, 3.30550753788332429831220007853, 3.39858832375687694422553744538, 3.52642814685450245174741473598, 3.56206085598588991731149537214, 3.66499593258028599044560419137, 3.71831926846712717981291729930, 4.21318395765081452597108142142, 4.31663946479509068026272547726, 4.35368256992059494133438665970, 4.45529678091106265882539483537
Plot not available for L-functions of degree greater than 10.
