| L(s) = 1 | − 2-s − 9·3-s + 11·5-s + 9·6-s + 21·8-s + 27·9-s − 11·10-s − 35·11-s − 124·13-s − 99·15-s − 31·16-s + 48·17-s − 27·18-s − 202·19-s + 35·22-s − 216·23-s − 189·24-s + 183·25-s + 124·26-s + 54·27-s + 106·29-s + 99·30-s − 95·31-s − 64·32-s + 315·33-s − 48·34-s − 262·37-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 1.73·3-s + 0.983·5-s + 0.612·6-s + 0.928·8-s + 9-s − 0.347·10-s − 0.959·11-s − 2.64·13-s − 1.70·15-s − 0.484·16-s + 0.684·17-s − 0.353·18-s − 2.43·19-s + 0.339·22-s − 1.95·23-s − 1.60·24-s + 1.46·25-s + 0.935·26-s + 0.384·27-s + 0.678·29-s + 0.602·30-s − 0.550·31-s − 0.353·32-s + 1.66·33-s − 0.242·34-s − 1.16·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2514833931\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2514833931\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( ( 1 + p T + p^{2} T^{2} )^{3} \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + T + T^{2} - 5 p^{2} T^{3} - 5 p T^{4} + p^{6} T^{5} + 265 p^{2} T^{6} + p^{9} T^{7} - 5 p^{7} T^{8} - 5 p^{11} T^{9} + p^{12} T^{10} + p^{15} T^{11} + p^{18} T^{12} \) |
| 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
−6.76870367834723335332825582149, −6.59516158174197404017160849702, −6.43321156440088790268761067141, −6.02457644356199206971535000419, −5.81502548410588056292781192190, −5.75154108682978443626818290439, −5.63934528025193744540060609714, −5.15060713154179506295315021222, −5.13066906948905476257307186330, −4.93582705870822397812164409094, −4.75163717551516524574216978547, −4.46828040578346355246335703039, −4.41919504632536102489123976200, −3.95061587395213310660861486420, −3.60714595349898151126985972359, −3.50680614255033585638659901208, −2.94694757304272117330287773144, −2.54066013098115001613200245043, −2.30210140327570921109045401688, −2.17335533144352096525626035939, −2.05059246795715220688861912839, −1.41367341242992595578418365154, −1.08665066379249628164541900338, −0.35351063831337415571864782813, −0.19496660691023901846797406598,
0.19496660691023901846797406598, 0.35351063831337415571864782813, 1.08665066379249628164541900338, 1.41367341242992595578418365154, 2.05059246795715220688861912839, 2.17335533144352096525626035939, 2.30210140327570921109045401688, 2.54066013098115001613200245043, 2.94694757304272117330287773144, 3.50680614255033585638659901208, 3.60714595349898151126985972359, 3.95061587395213310660861486420, 4.41919504632536102489123976200, 4.46828040578346355246335703039, 4.75163717551516524574216978547, 4.93582705870822397812164409094, 5.13066906948905476257307186330, 5.15060713154179506295315021222, 5.63934528025193744540060609714, 5.75154108682978443626818290439, 5.81502548410588056292781192190, 6.02457644356199206971535000419, 6.43321156440088790268761067141, 6.59516158174197404017160849702, 6.76870367834723335332825582149
Plot not available for L-functions of degree greater than 10.
