| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{18}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8109180723\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8109180723\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 93 T - 1689729 T^{2} + 354366248 T^{3} + 1310601420297 T^{4} - 224207143910091 T^{5} - 1053719651856288018 T^{6} - 224207143910091 p^{3} T^{7} + 1310601420297 p^{6} T^{8} + 354366248 p^{9} T^{9} - 1689729 p^{12} T^{10} - 93 p^{15} T^{11} + p^{18} T^{12} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.14747505049895992718645475207, −6.94385304599567988726099320141, −6.82786670829519917594680307417, −6.80310485693927456821688830641, −6.20517444479461668825884209700, −6.07135521324305211373277146049, −5.74365818668759514831955687625, −5.50951495613372034614665174932, −5.33381873979545944384191600159, −5.29664639856771086251717932999, −5.29545941711378763869726626505, −4.75352069789129608425317439024, −4.31112586165925539767884154710, −4.10051637422683345070103402478, −3.86032498225292129408747640846, −3.65371884616420700930501210720, −3.19849824177341612019070204320, −3.16690435528968200793987770574, −3.04706328400482937573617197524, −2.34826370692623216874775066081, −1.98419366211113976054550999618, −1.85921228953173603111288956228, −1.12322978972180077391822437651, −0.838075709161267440041902480326, −0.18563660148585931195432112021,
0.18563660148585931195432112021, 0.838075709161267440041902480326, 1.12322978972180077391822437651, 1.85921228953173603111288956228, 1.98419366211113976054550999618, 2.34826370692623216874775066081, 3.04706328400482937573617197524, 3.16690435528968200793987770574, 3.19849824177341612019070204320, 3.65371884616420700930501210720, 3.86032498225292129408747640846, 4.10051637422683345070103402478, 4.31112586165925539767884154710, 4.75352069789129608425317439024, 5.29545941711378763869726626505, 5.29664639856771086251717932999, 5.33381873979545944384191600159, 5.50951495613372034614665174932, 5.74365818668759514831955687625, 6.07135521324305211373277146049, 6.20517444479461668825884209700, 6.80310485693927456821688830641, 6.82786670829519917594680307417, 6.94385304599567988726099320141, 7.14747505049895992718645475207
Plot not available for L-functions of degree greater than 10.
