| L(s) = 1 | + 6·2-s + 12·3-s + 12·4-s + 24·5-s + 72·6-s + 27·7-s − 16·8-s + 108·9-s + 144·10-s − 82·11-s + 144·12-s + 162·14-s + 288·15-s − 144·16-s + 90·17-s + 648·18-s − 130·19-s + 288·20-s + 324·21-s − 492·22-s − 19·23-s − 192·24-s − 148·25-s + 890·27-s + 324·28-s + 101·29-s + 1.72e3·30-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 2.30·3-s + 3/2·4-s + 2.14·5-s + 4.89·6-s + 1.45·7-s − 0.707·8-s + 4·9-s + 4.55·10-s − 2.24·11-s + 3.46·12-s + 3.09·14-s + 4.95·15-s − 9/4·16-s + 1.28·17-s + 8.48·18-s − 1.56·19-s + 3.21·20-s + 3.36·21-s − 4.76·22-s − 0.172·23-s − 1.63·24-s − 1.18·25-s + 6.34·27-s + 2.18·28-s + 0.646·29-s + 10.5·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(75.83792900\) |
| \(L(\frac12)\) |
\(\approx\) |
\(75.83792900\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - p T + p^{2} T^{2} )^{3} \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 4 p T + 4 p^{2} T^{2} - 26 T^{3} + 376 p T^{4} - 272 p^{3} T^{5} + 22687 T^{6} - 272 p^{6} T^{7} + 376 p^{7} T^{8} - 26 p^{9} T^{9} + 4 p^{14} T^{10} - 4 p^{16} T^{11} + p^{18} T^{12} \) |
| 5 | \( ( 1 - 12 T + 58 p T^{2} - 2231 T^{3} + 58 p^{4} T^{4} - 12 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 7 | \( 1 - 27 T - 242 T^{2} + 1459 p T^{3} + 117983 T^{4} - 2878426 T^{5} + 3332855 T^{6} - 2878426 p^{3} T^{7} + 117983 p^{6} T^{8} + 1459 p^{10} T^{9} - 242 p^{12} T^{10} - 27 p^{15} T^{11} + p^{18} T^{12} \) |
| 11 | \( 1 + 82 T + 172 p T^{2} - 7922 T^{3} + 346846 T^{4} + 66936450 T^{5} + 2560930403 T^{6} + 66936450 p^{3} T^{7} + 346846 p^{6} T^{8} - 7922 p^{9} T^{9} + 172 p^{13} T^{10} + 82 p^{15} T^{11} + p^{18} T^{12} \) |
| 17 | \( 1 - 90 T - 5468 T^{2} + 393226 T^{3} + 44785470 T^{4} - 1125379354 T^{5} - 211247539689 T^{6} - 1125379354 p^{3} T^{7} + 44785470 p^{6} T^{8} + 393226 p^{9} T^{9} - 5468 p^{12} T^{10} - 90 p^{15} T^{11} + p^{18} T^{12} \) |
| 19 | \( 1 + 130 T + 492 T^{2} - 1280422 T^{3} - 65195038 T^{4} + 4858964058 T^{5} + 916633832443 T^{6} + 4858964058 p^{3} T^{7} - 65195038 p^{6} T^{8} - 1280422 p^{9} T^{9} + 492 p^{12} T^{10} + 130 p^{15} T^{11} + p^{18} T^{12} \) |
| 23 | \( 1 + 19 T - 17379 T^{2} - 1795946 T^{3} + 78601463 T^{4} + 13530427071 T^{5} + 857245646806 T^{6} + 13530427071 p^{3} T^{7} + 78601463 p^{6} T^{8} - 1795946 p^{9} T^{9} - 17379 p^{12} T^{10} + 19 p^{15} T^{11} + p^{18} T^{12} \) |
| 29 | \( 1 - 101 T + 9670 T^{2} + 2392827 T^{3} - 136214727 T^{4} - 58110677102 T^{5} + 24435911693829 T^{6} - 58110677102 p^{3} T^{7} - 136214727 p^{6} T^{8} + 2392827 p^{9} T^{9} + 9670 p^{12} T^{10} - 101 p^{15} T^{11} + p^{18} T^{12} \) |
| 31 | \( ( 1 - 519 T + 168527 T^{2} - 34325015 T^{3} + 168527 p^{3} T^{4} - 519 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 37 | \( 1 + 84 T - 141032 T^{2} - 4641854 T^{3} + 13711541948 T^{4} + 220099751228 T^{5} - 796061158324453 T^{6} + 220099751228 p^{3} T^{7} + 13711541948 p^{6} T^{8} - 4641854 p^{9} T^{9} - 141032 p^{12} T^{10} + 84 p^{15} T^{11} + p^{18} T^{12} \) |
| 41 | \( 1 + 187 T - 18745 T^{2} - 1163696 T^{3} - 1864796807 T^{4} - 195267953691 T^{5} + 321213047237726 T^{6} - 195267953691 p^{3} T^{7} - 1864796807 p^{6} T^{8} - 1163696 p^{9} T^{9} - 18745 p^{12} T^{10} + 187 p^{15} T^{11} + p^{18} T^{12} \) |
| 43 | \( 1 - 1205 T + 731136 T^{2} - 357194599 T^{3} + 154200549383 T^{4} - 53845297050762 T^{5} + 15910224576330043 T^{6} - 53845297050762 p^{3} T^{7} + 154200549383 p^{6} T^{8} - 357194599 p^{9} T^{9} + 731136 p^{12} T^{10} - 1205 p^{15} T^{11} + p^{18} T^{12} \) |
| 47 | \( ( 1 + 536 T + 236110 T^{2} + 85169985 T^{3} + 236110 p^{3} T^{4} + 536 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 53 | \( ( 1 + 1095 T + 839313 T^{2} + 371911379 T^{3} + 839313 p^{3} T^{4} + 1095 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 59 | \( 1 - 1413 T + 791881 T^{2} - 395784094 T^{3} + 4760265513 p T^{4} - 147325234570781 T^{5} + 61745012937499398 T^{6} - 147325234570781 p^{3} T^{7} + 4760265513 p^{7} T^{8} - 395784094 p^{9} T^{9} + 791881 p^{12} T^{10} - 1413 p^{15} T^{11} + p^{18} T^{12} \) |
| 61 | \( 1 - 1108 T + 195280 T^{2} - 81146578 T^{3} + 242730039476 T^{4} - 86381104403572 T^{5} + 569801340677051 T^{6} - 86381104403572 p^{3} T^{7} + 242730039476 p^{6} T^{8} - 81146578 p^{9} T^{9} + 195280 p^{12} T^{10} - 1108 p^{15} T^{11} + p^{18} T^{12} \) |
| 67 | \( 1 - 1605 T + 910660 T^{2} - 521034463 T^{3} + 545679629225 T^{4} - 329190135263360 T^{5} + 144884674066779755 T^{6} - 329190135263360 p^{3} T^{7} + 545679629225 p^{6} T^{8} - 521034463 p^{9} T^{9} + 910660 p^{12} T^{10} - 1605 p^{15} T^{11} + p^{18} T^{12} \) |
| 71 | \( 1 - 909 T - 188832 T^{2} + 363557137 T^{3} + 82513282683 T^{4} - 108794514090066 T^{5} + 30677817845635031 T^{6} - 108794514090066 p^{3} T^{7} + 82513282683 p^{6} T^{8} + 363557137 p^{9} T^{9} - 188832 p^{12} T^{10} - 909 p^{15} T^{11} + p^{18} T^{12} \) |
| 73 | \( ( 1 - 287 T + 330383 T^{2} - 44761521 T^{3} + 330383 p^{3} T^{4} - 287 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 79 | \( ( 1 + 1961 T + 2641773 T^{2} + 2147763857 T^{3} + 2641773 p^{3} T^{4} + 1961 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 83 | \( ( 1 - 191 T + 1581331 T^{2} - 188435781 T^{3} + 1581331 p^{3} T^{4} - 191 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 89 | \( 1 + 1091 T - 1221116 T^{2} - 467120403 T^{3} + 2424941163801 T^{4} + 568092710973530 T^{5} - 1590782774768604519 T^{6} + 568092710973530 p^{3} T^{7} + 2424941163801 p^{6} T^{8} - 467120403 p^{9} T^{9} - 1221116 p^{12} T^{10} + 1091 p^{15} T^{11} + p^{18} T^{12} \) |
| 97 | \( 1 - 947 T - 1184994 T^{2} + 1028052401 T^{3} + 1013625696203 T^{4} - 312418983925740 T^{5} - 932711772529957343 T^{6} - 312418983925740 p^{3} T^{7} + 1013625696203 p^{6} T^{8} + 1028052401 p^{9} T^{9} - 1184994 p^{12} T^{10} - 947 p^{15} T^{11} + p^{18} T^{12} \) |
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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
−5.99497966750363648610796196676, −5.33616711429272316411229581420, −5.29163182565577810640190693331, −5.09629228600917135600808059315, −4.98082721206099078522670748968, −4.92444598129373887976544924042, −4.85033077413676610820447171656, −4.37330058479914445743904206286, −4.12613321646576111564151737233, −4.05012858621967743065775819574, −4.03806535787713110397928422795, −3.78404999486496844735709459735, −3.59819301963575328488713784598, −2.99299971755671735902898490000, −2.77958193027309022730476705826, −2.70555808031958039760688698305, −2.60061189598415159541384975493, −2.52171916662755754108608889458, −2.37777608752468918341659120484, −1.92554445152553832971210116648, −1.59115118833607130762161677267, −1.45869956145610847792304300067, −1.18406414129596056281812996160, −0.814021371173117294748187586922, −0.29577942596693607327655230950,
0.29577942596693607327655230950, 0.814021371173117294748187586922, 1.18406414129596056281812996160, 1.45869956145610847792304300067, 1.59115118833607130762161677267, 1.92554445152553832971210116648, 2.37777608752468918341659120484, 2.52171916662755754108608889458, 2.60061189598415159541384975493, 2.70555808031958039760688698305, 2.77958193027309022730476705826, 2.99299971755671735902898490000, 3.59819301963575328488713784598, 3.78404999486496844735709459735, 4.03806535787713110397928422795, 4.05012858621967743065775819574, 4.12613321646576111564151737233, 4.37330058479914445743904206286, 4.85033077413676610820447171656, 4.92444598129373887976544924042, 4.98082721206099078522670748968, 5.09629228600917135600808059315, 5.29163182565577810640190693331, 5.33616711429272316411229581420, 5.99497966750363648610796196676
Plot not available for L-functions of degree greater than 10.
