| L(s) = 1 | − 6·5-s + 6·7-s + 51·11-s + 12·13-s + 222·17-s − 30·19-s + 210·23-s + 204·25-s − 456·29-s − 48·31-s − 36·35-s − 96·37-s − 897·41-s − 129·43-s + 522·47-s + 420·49-s + 2.20e3·53-s − 306·55-s + 453·59-s − 402·61-s − 72·65-s + 213·67-s + 120·71-s + 750·73-s + 306·77-s − 552·79-s − 612·83-s + ⋯ |
| L(s) = 1 | − 0.536·5-s + 0.323·7-s + 1.39·11-s + 0.256·13-s + 3.16·17-s − 0.362·19-s + 1.90·23-s + 1.63·25-s − 2.91·29-s − 0.278·31-s − 0.173·35-s − 0.426·37-s − 3.41·41-s − 0.457·43-s + 1.62·47-s + 1.22·49-s + 5.72·53-s − 0.750·55-s + 0.999·59-s − 0.843·61-s − 0.137·65-s + 0.388·67-s + 0.200·71-s + 1.20·73-s + 0.452·77-s − 0.786·79-s − 0.809·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{18}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(19.01414664\) |
| \(L(\frac12)\) |
\(\approx\) |
\(19.01414664\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 6 T - 168 T^{2} - 48 T^{3} + 492 p^{2} T^{4} - 82506 T^{5} - 1453754 T^{6} - 82506 p^{3} T^{7} + 492 p^{8} T^{8} - 48 p^{9} T^{9} - 168 p^{12} T^{10} + 6 p^{15} T^{11} + p^{18} T^{12} \) |
| 7 | \( 1 - 6 T - 384 T^{2} + 14908 T^{3} - 19944 T^{4} - 376722 p T^{5} + 91900446 T^{6} - 376722 p^{4} T^{7} - 19944 p^{6} T^{8} + 14908 p^{9} T^{9} - 384 p^{12} T^{10} - 6 p^{15} T^{11} + p^{18} T^{12} \) |
| 11 | \( 1 - 51 T - 195 T^{2} - 3534 T^{3} + 550059 T^{4} + 8621115 p T^{5} - 45145634 p^{2} T^{6} + 8621115 p^{4} T^{7} + 550059 p^{6} T^{8} - 3534 p^{9} T^{9} - 195 p^{12} T^{10} - 51 p^{15} T^{11} + p^{18} T^{12} \) |
| 13 | \( 1 - 12 T - 1176 T^{2} - 39316 T^{3} - 1016784 T^{4} + 27173412 T^{5} + 16823666094 T^{6} + 27173412 p^{3} T^{7} - 1016784 p^{6} T^{8} - 39316 p^{9} T^{9} - 1176 p^{12} T^{10} - 12 p^{15} T^{11} + p^{18} T^{12} \) |
| 17 | \( ( 1 - 111 T + 8115 T^{2} - 513210 T^{3} + 8115 p^{3} T^{4} - 111 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 19 | \( ( 1 + 15 T + 13065 T^{2} + 422138 T^{3} + 13065 p^{3} T^{4} + 15 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 23 | \( 1 - 210 T - 120 p T^{2} + 384324 T^{3} + 552096960 T^{4} - 31196345610 T^{5} - 3505833433730 T^{6} - 31196345610 p^{3} T^{7} + 552096960 p^{6} T^{8} + 384324 p^{9} T^{9} - 120 p^{13} T^{10} - 210 p^{15} T^{11} + p^{18} T^{12} \) |
| 29 | \( 1 + 456 T + 70032 T^{2} + 12639372 T^{3} + 4198346304 T^{4} + 668355791208 T^{5} + 72647268176734 T^{6} + 668355791208 p^{3} T^{7} + 4198346304 p^{6} T^{8} + 12639372 p^{9} T^{9} + 70032 p^{12} T^{10} + 456 p^{15} T^{11} + p^{18} T^{12} \) |
| 31 | \( 1 + 48 T - 52116 T^{2} + 3001864 T^{3} + 1349663076 T^{4} - 123641386272 T^{5} - 37354727750178 T^{6} - 123641386272 p^{3} T^{7} + 1349663076 p^{6} T^{8} + 3001864 p^{9} T^{9} - 52116 p^{12} T^{10} + 48 p^{15} T^{11} + p^{18} T^{12} \) |
| 37 | \( ( 1 + 48 T + 131451 T^{2} + 4180336 T^{3} + 131451 p^{3} T^{4} + 48 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 41 | \( 1 + 897 T + 374295 T^{2} + 115110024 T^{3} + 34022054553 T^{4} + 9337061108679 T^{5} + 2414891032993726 T^{6} + 9337061108679 p^{3} T^{7} + 34022054553 p^{6} T^{8} + 115110024 p^{9} T^{9} + 374295 p^{12} T^{10} + 897 p^{15} T^{11} + p^{18} T^{12} \) |
| 43 | \( 1 + 3 p T - 34971 T^{2} - 732962 p T^{3} - 3902024493 T^{4} + 17253512631 p T^{5} + 569247939582 p^{2} T^{6} + 17253512631 p^{4} T^{7} - 3902024493 p^{6} T^{8} - 732962 p^{10} T^{9} - 34971 p^{12} T^{10} + 3 p^{16} T^{11} + p^{18} T^{12} \) |
| 47 | \( 1 - 522 T + 122448 T^{2} + 9346068 T^{3} - 15946914792 T^{4} + 4059391014414 T^{5} - 1114759766751410 T^{6} + 4059391014414 p^{3} T^{7} - 15946914792 p^{6} T^{8} + 9346068 p^{9} T^{9} + 122448 p^{12} T^{10} - 522 p^{15} T^{11} + p^{18} T^{12} \) |
| 53 | \( ( 1 - 1104 T + 764331 T^{2} - 340574064 T^{3} + 764331 p^{3} T^{4} - 1104 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 59 | \( 1 - 453 T - 119643 T^{2} + 31203366 T^{3} - 1587796437 T^{4} + 12368009864103 T^{5} - 5050047108050786 T^{6} + 12368009864103 p^{3} T^{7} - 1587796437 p^{6} T^{8} + 31203366 p^{9} T^{9} - 119643 p^{12} T^{10} - 453 p^{15} T^{11} + p^{18} T^{12} \) |
| 61 | \( 1 + 402 T - 527280 T^{2} - 85634608 T^{3} + 245321825076 T^{4} + 19877822718498 T^{5} - 59790452550726954 T^{6} + 19877822718498 p^{3} T^{7} + 245321825076 p^{6} T^{8} - 85634608 p^{9} T^{9} - 527280 p^{12} T^{10} + 402 p^{15} T^{11} + p^{18} T^{12} \) |
| 67 | \( 1 - 213 T - 776787 T^{2} + 87320974 T^{3} + 400716501795 T^{4} - 21812260017825 T^{5} - 135770111734985634 T^{6} - 21812260017825 p^{3} T^{7} + 400716501795 p^{6} T^{8} + 87320974 p^{9} T^{9} - 776787 p^{12} T^{10} - 213 p^{15} T^{11} + p^{18} T^{12} \) |
| 71 | \( ( 1 - 60 T + 650661 T^{2} + 70262328 T^{3} + 650661 p^{3} T^{4} - 60 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 73 | \( ( 1 - 375 T + 786003 T^{2} - 133393466 T^{3} + 786003 p^{3} T^{4} - 375 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 79 | \( 1 + 552 T - 1152564 T^{2} - 248520632 T^{3} + 1114530677604 T^{4} + 108173831172696 T^{5} - 604232203209226626 T^{6} + 108173831172696 p^{3} T^{7} + 1114530677604 p^{6} T^{8} - 248520632 p^{9} T^{9} - 1152564 p^{12} T^{10} + 552 p^{15} T^{11} + p^{18} T^{12} \) |
| 83 | \( 1 + 612 T - 1223124 T^{2} - 415004184 T^{3} + 1212827483508 T^{4} + 194529560235444 T^{5} - 723127027932774218 T^{6} + 194529560235444 p^{3} T^{7} + 1212827483508 p^{6} T^{8} - 415004184 p^{9} T^{9} - 1223124 p^{12} T^{10} + 612 p^{15} T^{11} + p^{18} T^{12} \) |
| 89 | \( ( 1 - 462 T + 1340727 T^{2} - 821513604 T^{3} + 1340727 p^{3} T^{4} - 462 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 97 | \( 1 - 93 T - 1689729 T^{2} + 354366248 T^{3} + 1310601420297 T^{4} - 224207143910091 T^{5} - 1053719651856288018 T^{6} - 224207143910091 p^{3} T^{7} + 1310601420297 p^{6} T^{8} + 354366248 p^{9} T^{9} - 1689729 p^{12} T^{10} - 93 p^{15} T^{11} + p^{18} T^{12} \) |
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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
−5.55997900315628847411395823758, −5.36590699786218827201142200309, −5.31952230396786253254015676131, −5.21309041669300756428632695391, −4.97232425510854870651248075460, −4.60355272866450698454612507944, −4.35888272961016239601942745539, −4.29472883958698800722248547966, −3.90391361374538676016876685429, −3.72457717035284500132820565104, −3.63871941325803115186987531067, −3.61841275277353791608314136765, −3.54702745411424664082801721628, −2.92258370283736793171487750836, −2.91386058460510839239441432550, −2.70723383414267774836789762096, −2.50855212387265743601596999697, −1.88521954042354041232958766442, −1.82700293904237494811054717979, −1.60704711569694752369754695301, −1.46057386210535738927917856940, −0.907457804496342490142682764621, −0.77714758427039338474854648909, −0.59775540494914384027184748725, −0.49135726252674069610118451686,
0.49135726252674069610118451686, 0.59775540494914384027184748725, 0.77714758427039338474854648909, 0.907457804496342490142682764621, 1.46057386210535738927917856940, 1.60704711569694752369754695301, 1.82700293904237494811054717979, 1.88521954042354041232958766442, 2.50855212387265743601596999697, 2.70723383414267774836789762096, 2.91386058460510839239441432550, 2.92258370283736793171487750836, 3.54702745411424664082801721628, 3.61841275277353791608314136765, 3.63871941325803115186987531067, 3.72457717035284500132820565104, 3.90391361374538676016876685429, 4.29472883958698800722248547966, 4.35888272961016239601942745539, 4.60355272866450698454612507944, 4.97232425510854870651248075460, 5.21309041669300756428632695391, 5.31952230396786253254015676131, 5.36590699786218827201142200309, 5.55997900315628847411395823758
Plot not available for L-functions of degree greater than 10.
