| L(s) = 1 | − 9·3-s − 11·5-s + 13·7-s + 27·9-s + 35·11-s + 124·13-s + 99·15-s − 48·17-s − 202·19-s − 117·21-s + 216·23-s + 183·25-s + 54·27-s + 106·29-s − 95·31-s − 315·33-s − 143·35-s − 262·37-s − 1.11e3·39-s + 488·41-s − 720·43-s − 297·45-s − 210·47-s − 67·49-s + 432·51-s − 393·53-s − 385·55-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.983·5-s + 0.701·7-s + 9-s + 0.959·11-s + 2.64·13-s + 1.70·15-s − 0.684·17-s − 2.43·19-s − 1.21·21-s + 1.95·23-s + 1.46·25-s + 0.384·27-s + 0.678·29-s − 0.550·31-s − 1.66·33-s − 0.690·35-s − 1.16·37-s − 4.58·39-s + 1.85·41-s − 2.55·43-s − 0.983·45-s − 0.651·47-s − 0.195·49-s + 1.18·51-s − 1.01·53-s − 0.943·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1105612109\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1105612109\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + p T + p^{2} T^{2} )^{3} \) |
| 7 | \( 1 - 13 T + 236 T^{2} - 1735 p T^{3} + 236 p^{3} T^{4} - 13 p^{6} T^{5} + p^{9} T^{6} \) |
| good | 5 | \( 1 + 11 T - 62 T^{2} - 203 p T^{3} - 1208 p T^{4} - 54313 T^{5} + 121696 T^{6} - 54313 p^{3} T^{7} - 1208 p^{7} T^{8} - 203 p^{10} T^{9} - 62 p^{12} T^{10} + 11 p^{15} T^{11} + p^{18} T^{12} \) |
| 11 | \( 1 - 35 T - 1400 T^{2} + 113593 T^{3} - 198940 T^{4} - 87110135 T^{5} + 3928586038 T^{6} - 87110135 p^{3} T^{7} - 198940 p^{6} T^{8} + 113593 p^{9} T^{9} - 1400 p^{12} T^{10} - 35 p^{15} T^{11} + p^{18} T^{12} \) |
| 13 | \( ( 1 - 62 T + 7016 T^{2} - 253976 T^{3} + 7016 p^{3} T^{4} - 62 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 17 | \( 1 + 48 T - 10035 T^{2} - 125232 T^{3} + 74409318 T^{4} - 234420432 T^{5} - 437742983351 T^{6} - 234420432 p^{3} T^{7} + 74409318 p^{6} T^{8} - 125232 p^{9} T^{9} - 10035 p^{12} T^{10} + 48 p^{15} T^{11} + p^{18} T^{12} \) |
| 19 | \( 1 + 202 T + 7946 T^{2} + 627636 T^{3} + 247297462 T^{4} + 17185599794 T^{5} + 349471935958 T^{6} + 17185599794 p^{3} T^{7} + 247297462 p^{6} T^{8} + 627636 p^{9} T^{9} + 7946 p^{12} T^{10} + 202 p^{15} T^{11} + p^{18} T^{12} \) |
| 23 | \( 1 - 216 T + 10827 T^{2} - 387864 T^{3} + 53856198 T^{4} + 24653558952 T^{5} - 5413409425505 T^{6} + 24653558952 p^{3} T^{7} + 53856198 p^{6} T^{8} - 387864 p^{9} T^{9} + 10827 p^{12} T^{10} - 216 p^{15} T^{11} + p^{18} T^{12} \) |
| 29 | \( ( 1 - 53 T + 52695 T^{2} - 3410210 T^{3} + 52695 p^{3} T^{4} - 53 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 31 | \( 1 + 95 T - 70347 T^{2} - 3756594 T^{3} + 3398738767 T^{4} + 83374434539 T^{5} - 110906046363338 T^{6} + 83374434539 p^{3} T^{7} + 3398738767 p^{6} T^{8} - 3756594 p^{9} T^{9} - 70347 p^{12} T^{10} + 95 p^{15} T^{11} + p^{18} T^{12} \) |
| 37 | \( 1 + 262 T - 97404 T^{2} - 9678072 T^{3} + 12194182072 T^{4} + 680381910454 T^{5} - 605701122868778 T^{6} + 680381910454 p^{3} T^{7} + 12194182072 p^{6} T^{8} - 9678072 p^{9} T^{9} - 97404 p^{12} T^{10} + 262 p^{15} T^{11} + p^{18} T^{12} \) |
| 41 | \( ( 1 - 244 T + 187983 T^{2} - 33933832 T^{3} + 187983 p^{3} T^{4} - 244 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 43 | \( ( 1 + 360 T + 166158 T^{2} + 38975294 T^{3} + 166158 p^{3} T^{4} + 360 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 47 | \( 1 + 210 T - 20853 T^{2} - 83809446 T^{3} - 12756928590 T^{4} + 2596137940074 T^{5} + 3698984470026571 T^{6} + 2596137940074 p^{3} T^{7} - 12756928590 p^{6} T^{8} - 83809446 p^{9} T^{9} - 20853 p^{12} T^{10} + 210 p^{15} T^{11} + p^{18} T^{12} \) |
| 53 | \( 1 + 393 T - 211446 T^{2} - 23899125 T^{3} + 46453564620 T^{4} - 3425920762143 T^{5} - 9724787230272680 T^{6} - 3425920762143 p^{3} T^{7} + 46453564620 p^{6} T^{8} - 23899125 p^{9} T^{9} - 211446 p^{12} T^{10} + 393 p^{15} T^{11} + p^{18} T^{12} \) |
| 59 | \( 1 - 1143 T + 557208 T^{2} - 118327563 T^{3} - 14314666608 T^{4} + 458696646099 p T^{5} - 16891447327378130 T^{6} + 458696646099 p^{4} T^{7} - 14314666608 p^{6} T^{8} - 118327563 p^{9} T^{9} + 557208 p^{12} T^{10} - 1143 p^{15} T^{11} + p^{18} T^{12} \) |
| 61 | \( 1 - 70 T - 335143 T^{2} - 129510330 T^{3} + 42145697866 T^{4} + 25171752927730 T^{5} - 316289217432887 T^{6} + 25171752927730 p^{3} T^{7} + 42145697866 p^{6} T^{8} - 129510330 p^{9} T^{9} - 335143 p^{12} T^{10} - 70 p^{15} T^{11} + p^{18} T^{12} \) |
| 67 | \( 1 + 628 T - 202942 T^{2} - 436381932 T^{3} - 77667044702 T^{4} + 1097444953952 p T^{5} + 16943668917790 p^{2} T^{6} + 1097444953952 p^{4} T^{7} - 77667044702 p^{6} T^{8} - 436381932 p^{9} T^{9} - 202942 p^{12} T^{10} + 628 p^{15} T^{11} + p^{18} T^{12} \) |
| 71 | \( ( 1 + 318 T + 742929 T^{2} + 256167372 T^{3} + 742929 p^{3} T^{4} + 318 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 73 | \( 1 + 988 T - 186552 T^{2} - 102237300 T^{3} + 281568890272 T^{4} - 16988127696596 T^{5} - 164639785652996186 T^{6} - 16988127696596 p^{3} T^{7} + 281568890272 p^{6} T^{8} - 102237300 p^{9} T^{9} - 186552 p^{12} T^{10} + 988 p^{15} T^{11} + p^{18} T^{12} \) |
| 79 | \( 1 - 861 T - 479895 T^{2} + 258646666 T^{3} + 325257480351 T^{4} + 27564282842211 T^{5} - 246706047980056146 T^{6} + 27564282842211 p^{3} T^{7} + 325257480351 p^{6} T^{8} + 258646666 p^{9} T^{9} - 479895 p^{12} T^{10} - 861 p^{15} T^{11} + p^{18} T^{12} \) |
| 83 | \( ( 1 + 519 T + 1583745 T^{2} + 545598870 T^{3} + 1583745 p^{3} T^{4} + 519 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 89 | \( 1 + 1766 T + 725929 T^{2} - 728159446 T^{3} - 335534377858 T^{4} + 846551335831238 T^{5} + 1249625385561159997 T^{6} + 846551335831238 p^{3} T^{7} - 335534377858 p^{6} T^{8} - 728159446 p^{9} T^{9} + 725929 p^{12} T^{10} + 1766 p^{15} T^{11} + p^{18} T^{12} \) |
| 97 | \( ( 1 - 19 T + 2168419 T^{2} + 10094878 T^{3} + 2168419 p^{3} T^{4} - 19 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
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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
−5.92422547671456639206689908385, −5.70040558023288124257786437286, −5.42464831209032576962435589618, −5.12912404910969385854759179472, −5.07946721374851089012198116396, −4.84075060893001106812472135075, −4.83189731738616839282600616649, −4.51270985189381862652062854615, −4.16926607497905593519412166345, −4.14325044447980135400695694148, −4.00975003014972250394910976009, −3.77923173955743028313161391162, −3.51234074273613870301861129901, −3.23400973665991076149526686226, −3.08303153134362467620373442048, −2.89157576292541202885501124459, −2.35665490359158363450495122272, −2.30491151276552549928641510857, −1.95340454578621793811353057383, −1.30712247081533675948603369989, −1.22055465967484617982988449749, −1.20296048583538723598368712157, −1.14645049960834012591034312431, −0.25730605606593441529918929899, −0.088990211620743331542927790705,
0.088990211620743331542927790705, 0.25730605606593441529918929899, 1.14645049960834012591034312431, 1.20296048583538723598368712157, 1.22055465967484617982988449749, 1.30712247081533675948603369989, 1.95340454578621793811353057383, 2.30491151276552549928641510857, 2.35665490359158363450495122272, 2.89157576292541202885501124459, 3.08303153134362467620373442048, 3.23400973665991076149526686226, 3.51234074273613870301861129901, 3.77923173955743028313161391162, 4.00975003014972250394910976009, 4.14325044447980135400695694148, 4.16926607497905593519412166345, 4.51270985189381862652062854615, 4.83189731738616839282600616649, 4.84075060893001106812472135075, 5.07946721374851089012198116396, 5.12912404910969385854759179472, 5.42464831209032576962435589618, 5.70040558023288124257786437286, 5.92422547671456639206689908385
Plot not available for L-functions of degree greater than 10.
