| L(s) = 1 | − 6·3-s − 7·4-s + 15·9-s + 42·12-s + 26·16-s − 14·17-s + 4·23-s + 40·25-s − 14·27-s + 16·29-s − 105·36-s − 6·43-s − 156·48-s − 4·49-s + 84·51-s − 52·53-s + 26·61-s − 77·64-s + 98·68-s − 24·69-s − 240·75-s − 36·79-s − 21·81-s − 96·87-s − 28·92-s − 280·100-s − 8·103-s + ⋯ |
| L(s) = 1 | − 3.46·3-s − 7/2·4-s + 5·9-s + 12.1·12-s + 13/2·16-s − 3.39·17-s + 0.834·23-s + 8·25-s − 2.69·27-s + 2.97·29-s − 17.5·36-s − 0.914·43-s − 22.5·48-s − 4/7·49-s + 11.7·51-s − 7.14·53-s + 3.32·61-s − 9.62·64-s + 11.8·68-s − 2.88·69-s − 27.7·75-s − 4.05·79-s − 7/3·81-s − 10.2·87-s − 2.91·92-s − 28·100-s − 0.788·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 13^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 13^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2882100224\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2882100224\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( ( 1 + T + T^{2} )^{6} \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 7 T^{2} + 23 T^{4} + 7 p^{3} T^{6} + 131 T^{8} + 147 p T^{10} + 609 T^{12} + 147 p^{3} T^{14} + 131 p^{4} T^{16} + 7 p^{9} T^{18} + 23 p^{8} T^{20} + 7 p^{10} T^{22} + p^{12} T^{24} \) |
| 5 | \( ( 1 - 4 p T^{2} + 192 T^{4} - 1169 T^{6} + 192 p^{2} T^{8} - 4 p^{5} T^{10} + p^{6} T^{12} )^{2} \) |
| 7 | \( 1 + 4 T^{2} - 36 T^{4} - 22 p T^{6} - 568 T^{8} - 1368 T^{10} + 97147 T^{12} - 1368 p^{2} T^{14} - 568 p^{4} T^{16} - 22 p^{7} T^{18} - 36 p^{8} T^{20} + 4 p^{10} T^{22} + p^{12} T^{24} \) |
| 11 | \( 1 + 5 T^{2} - 92 T^{4} - 3769 T^{6} - 11353 T^{8} + 186344 T^{10} + 6260057 T^{12} + 186344 p^{2} T^{14} - 11353 p^{4} T^{16} - 3769 p^{6} T^{18} - 92 p^{8} T^{20} + 5 p^{10} T^{22} + p^{12} T^{24} \) |
| 17 | \( ( 1 + 7 T - 16 T^{2} - 35 T^{3} + 1405 T^{4} + 2478 T^{5} - 16135 T^{6} + 2478 p T^{7} + 1405 p^{2} T^{8} - 35 p^{3} T^{9} - 16 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 19 | \( 1 + 13 T^{2} - 156 T^{4} - 3601 T^{6} - 62569 T^{8} - 84552 T^{10} + 49248457 T^{12} - 84552 p^{2} T^{14} - 62569 p^{4} T^{16} - 3601 p^{6} T^{18} - 156 p^{8} T^{20} + 13 p^{10} T^{22} + p^{12} T^{24} \) |
| 23 | \( ( 1 - 2 T - 22 T^{2} + 298 T^{3} - 272 T^{4} - 3216 T^{5} + 37295 T^{6} - 3216 p T^{7} - 272 p^{2} T^{8} + 298 p^{3} T^{9} - 22 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 29 | \( ( 1 - 8 T - 28 T^{2} + 106 T^{3} + 2428 T^{4} - 2772 T^{5} - 73261 T^{6} - 2772 p T^{7} + 2428 p^{2} T^{8} + 106 p^{3} T^{9} - 28 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 31 | \( ( 1 - 76 T^{2} + 4456 T^{4} - 150973 T^{6} + 4456 p^{2} T^{8} - 76 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 37 | \( 1 + 152 T^{2} + 11508 T^{4} + 692686 T^{6} + 37279436 T^{8} + 1662210564 T^{10} + 63919930579 T^{12} + 1662210564 p^{2} T^{14} + 37279436 p^{4} T^{16} + 692686 p^{6} T^{18} + 11508 p^{8} T^{20} + 152 p^{10} T^{22} + p^{12} T^{24} \) |
| 41 | \( 1 + 241 T^{2} + 33680 T^{4} + 3223679 T^{6} + 234226799 T^{8} + 13326558240 T^{10} + 609356756721 T^{12} + 13326558240 p^{2} T^{14} + 234226799 p^{4} T^{16} + 3223679 p^{6} T^{18} + 33680 p^{8} T^{20} + 241 p^{10} T^{22} + p^{12} T^{24} \) |
| 43 | \( ( 1 + 3 T - 95 T^{2} - 262 T^{3} + 5483 T^{4} + 8563 T^{5} - 240346 T^{6} + 8563 p T^{7} + 5483 p^{2} T^{8} - 262 p^{3} T^{9} - 95 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 47 | \( ( 1 + 39 T^{2} + 3585 T^{4} + 112983 T^{6} + 3585 p^{2} T^{8} + 39 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 53 | \( ( 1 + 13 T + 199 T^{2} + 1407 T^{3} + 199 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 59 | \( 1 + 158 T^{2} + 17437 T^{4} + 797642 T^{6} - 484054 T^{8} - 4891249930 T^{10} - 377012986027 T^{12} - 4891249930 p^{2} T^{14} - 484054 p^{4} T^{16} + 797642 p^{6} T^{18} + 17437 p^{8} T^{20} + 158 p^{10} T^{22} + p^{12} T^{24} \) |
| 61 | \( ( 1 - 13 T - 26 T^{2} + 191 T^{3} + 8411 T^{4} - 4888 T^{5} - 636643 T^{6} - 4888 p T^{7} + 8411 p^{2} T^{8} + 191 p^{3} T^{9} - 26 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 67 | \( 1 + 333 T^{2} + 60592 T^{4} + 8064419 T^{6} + 858038723 T^{8} + 74740385692 T^{10} + 5445216739649 T^{12} + 74740385692 p^{2} T^{14} + 858038723 p^{4} T^{16} + 8064419 p^{6} T^{18} + 60592 p^{8} T^{20} + 333 p^{10} T^{22} + p^{12} T^{24} \) |
| 71 | \( 1 + 232 T^{2} + 21532 T^{4} + 1992262 T^{6} + 242049380 T^{8} + 19158383356 T^{10} + 1192897792955 T^{12} + 19158383356 p^{2} T^{14} + 242049380 p^{4} T^{16} + 1992262 p^{6} T^{18} + 21532 p^{8} T^{20} + 232 p^{10} T^{22} + p^{12} T^{24} \) |
| 73 | \( ( 1 - 316 T^{2} + 48500 T^{4} - 4463217 T^{6} + 48500 p^{2} T^{8} - 316 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 79 | \( ( 1 + 9 T + 215 T^{2} + 1253 T^{3} + 215 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 83 | \( ( 1 - 352 T^{2} + 59264 T^{4} - 6129693 T^{6} + 59264 p^{2} T^{8} - 352 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 89 | \( 1 + 493 T^{2} + 138776 T^{4} + 27085739 T^{6} + 4051822571 T^{8} + 484552737900 T^{10} + 47496547421529 T^{12} + 484552737900 p^{2} T^{14} + 4051822571 p^{4} T^{16} + 27085739 p^{6} T^{18} + 138776 p^{8} T^{20} + 493 p^{10} T^{22} + p^{12} T^{24} \) |
| 97 | \( 1 + 219 T^{2} + 7219 T^{4} + 533636 T^{6} + 357988697 T^{8} + 22373852977 T^{10} - 604855919098 T^{12} + 22373852977 p^{2} T^{14} + 357988697 p^{4} T^{16} + 533636 p^{6} T^{18} + 7219 p^{8} T^{20} + 219 p^{10} T^{22} + p^{12} T^{24} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.77056661561352096860819919455, −3.53272781393754365451363220863, −3.23334605440969161790327674890, −3.19717104586340876490429707256, −3.16830665626593588014925429340, −3.14992485777983304672731293756, −3.14029745802830822169776279500, −2.85119940877162239821246398307, −2.79909527343514097926599732398, −2.78245625435021061829802838446, −2.63290388025957239082632648528, −2.63226929964739974845627542132, −2.33599960684082878512451041543, −1.98590194505190117031276684309, −1.80772737857693240126084636458, −1.75015005353537646477471910523, −1.69287772818594812639172215163, −1.47016503720369968930984849513, −1.13256160462137661469720063768, −1.08942768258099858910050036241, −0.960492078174909365181795592660, −0.77919561434232622040616540013, −0.51356859656640151804959839816, −0.37034449921119987946255438498, −0.32504629097272838986612676365,
0.32504629097272838986612676365, 0.37034449921119987946255438498, 0.51356859656640151804959839816, 0.77919561434232622040616540013, 0.960492078174909365181795592660, 1.08942768258099858910050036241, 1.13256160462137661469720063768, 1.47016503720369968930984849513, 1.69287772818594812639172215163, 1.75015005353537646477471910523, 1.80772737857693240126084636458, 1.98590194505190117031276684309, 2.33599960684082878512451041543, 2.63226929964739974845627542132, 2.63290388025957239082632648528, 2.78245625435021061829802838446, 2.79909527343514097926599732398, 2.85119940877162239821246398307, 3.14029745802830822169776279500, 3.14992485777983304672731293756, 3.16830665626593588014925429340, 3.19717104586340876490429707256, 3.23334605440969161790327674890, 3.53272781393754365451363220863, 3.77056661561352096860819919455
Plot not available for L-functions of degree greater than 10.
