| L(s) = 1 | + 6·3-s + 8-s + 21·9-s + 3·11-s + 15·13-s + 6·17-s + 6·19-s + 15·23-s + 6·24-s + 55·27-s + 24·29-s − 6·31-s + 18·33-s − 30·37-s + 90·39-s + 18·41-s − 3·43-s − 3·47-s + 18·49-s + 36·51-s + 12·53-s + 36·57-s + 21·59-s − 21·61-s − 9·67-s + 90·69-s + 15·71-s + ⋯ |
| L(s) = 1 | + 3.46·3-s + 0.353·8-s + 7·9-s + 0.904·11-s + 4.16·13-s + 1.45·17-s + 1.37·19-s + 3.12·23-s + 1.22·24-s + 10.5·27-s + 4.45·29-s − 1.07·31-s + 3.13·33-s − 4.93·37-s + 14.4·39-s + 2.81·41-s − 0.457·43-s − 0.437·47-s + 18/7·49-s + 5.04·51-s + 1.64·53-s + 4.76·57-s + 2.73·59-s − 2.68·61-s − 1.09·67-s + 10.8·69-s + 1.78·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(71.88762769\) |
| \(L(\frac12)\) |
\(\approx\) |
\(71.88762769\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T^{3} + T^{6} \) |
| 5 | \( 1 \) |
| 19 | \( 1 - 6 T - 12 T^{2} + 169 T^{3} - 12 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| good | 3 | \( 1 - 2 p T + 5 p T^{2} - 19 T^{3} + p T^{4} + 17 p T^{5} - 134 T^{6} + 17 p^{2} T^{7} + p^{3} T^{8} - 19 p^{3} T^{9} + 5 p^{5} T^{10} - 2 p^{6} T^{11} + p^{6} T^{12} \) |
| 7 | \( 1 - 18 T^{2} - 2 T^{3} + 198 T^{4} + 18 T^{5} - 1581 T^{6} + 18 p T^{7} + 198 p^{2} T^{8} - 2 p^{3} T^{9} - 18 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( 1 - 3 T - 6 T^{2} + 81 T^{3} - 129 T^{4} - 318 T^{5} + 3067 T^{6} - 318 p T^{7} - 129 p^{2} T^{8} + 81 p^{3} T^{9} - 6 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 13 | \( 1 - 15 T + 120 T^{2} - 698 T^{3} + 3303 T^{4} - 13401 T^{5} + 49605 T^{6} - 13401 p T^{7} + 3303 p^{2} T^{8} - 698 p^{3} T^{9} + 120 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \) |
| 17 | \( 1 - 6 T + 27 T^{2} - 207 T^{3} + 801 T^{4} - 219 p T^{5} + 1286 p T^{6} - 219 p^{2} T^{7} + 801 p^{2} T^{8} - 207 p^{3} T^{9} + 27 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 23 | \( 1 - 15 T + 90 T^{2} - 162 T^{3} - 1017 T^{4} + 10101 T^{5} - 54485 T^{6} + 10101 p T^{7} - 1017 p^{2} T^{8} - 162 p^{3} T^{9} + 90 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \) |
| 29 | \( 1 - 24 T + 252 T^{2} - 1494 T^{3} + 4356 T^{4} + 8364 T^{5} - 131345 T^{6} + 8364 p T^{7} + 4356 p^{2} T^{8} - 1494 p^{3} T^{9} + 252 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \) |
| 31 | \( 1 + 6 T - 42 T^{2} - 130 T^{3} + 1872 T^{4} + 72 T^{5} - 74217 T^{6} + 72 p T^{7} + 1872 p^{2} T^{8} - 130 p^{3} T^{9} - 42 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 37 | \( ( 1 + 15 T + 150 T^{2} + 983 T^{3} + 150 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 41 | \( 1 - 18 T + 144 T^{2} - 720 T^{3} + 4068 T^{4} - 31752 T^{5} + 224083 T^{6} - 31752 p T^{7} + 4068 p^{2} T^{8} - 720 p^{3} T^{9} + 144 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \) |
| 43 | \( 1 + 3 T + 3 T^{2} + 121 T^{3} - 252 T^{4} - 10710 T^{5} - 79635 T^{6} - 10710 p T^{7} - 252 p^{2} T^{8} + 121 p^{3} T^{9} + 3 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 47 | \( 1 + 3 T + 36 T^{2} + 648 T^{3} + 3330 T^{4} + 19857 T^{5} + 286237 T^{6} + 19857 p T^{7} + 3330 p^{2} T^{8} + 648 p^{3} T^{9} + 36 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 53 | \( 1 - 12 T + 54 T^{2} + 27 T^{3} - 171 T^{4} - 18363 T^{5} + 156889 T^{6} - 18363 p T^{7} - 171 p^{2} T^{8} + 27 p^{3} T^{9} + 54 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 59 | \( 1 - 21 T + 81 T^{2} + 1647 T^{3} - 17208 T^{4} - 28362 T^{5} + 1122445 T^{6} - 28362 p T^{7} - 17208 p^{2} T^{8} + 1647 p^{3} T^{9} + 81 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \) |
| 61 | \( 1 + 21 T + 348 T^{2} + 4526 T^{3} + 50787 T^{4} + 478683 T^{5} + 3989277 T^{6} + 478683 p T^{7} + 50787 p^{2} T^{8} + 4526 p^{3} T^{9} + 348 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \) |
| 67 | \( 1 + 9 T - 18 T^{2} - 956 T^{3} - 6723 T^{4} + 42849 T^{5} + 795909 T^{6} + 42849 p T^{7} - 6723 p^{2} T^{8} - 956 p^{3} T^{9} - 18 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 71 | \( 1 - 15 T + 252 T^{2} - 2682 T^{3} + 27171 T^{4} - 243771 T^{5} + 1964377 T^{6} - 243771 p T^{7} + 27171 p^{2} T^{8} - 2682 p^{3} T^{9} + 252 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \) |
| 73 | \( 1 + 21 T + 183 T^{2} + 985 T^{3} + 1242 T^{4} - 12420 T^{5} + 19041 T^{6} - 12420 p T^{7} + 1242 p^{2} T^{8} + 985 p^{3} T^{9} + 183 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \) |
| 79 | \( 1 + 236 T^{3} - 437343 T^{6} + 236 p^{3} T^{9} + p^{6} T^{12} \) |
| 83 | \( 1 - 3 T - 186 T^{2} + 513 T^{3} + 20193 T^{4} - 30612 T^{5} - 1773101 T^{6} - 30612 p T^{7} + 20193 p^{2} T^{8} + 513 p^{3} T^{9} - 186 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 89 | \( 1 + 24 T + 234 T^{2} + 261 T^{3} - 11691 T^{4} - 1437 p T^{5} - 9991 p T^{6} - 1437 p^{2} T^{7} - 11691 p^{2} T^{8} + 261 p^{3} T^{9} + 234 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \) |
| 97 | \( 1 + 9 T + 306 T^{2} + 1330 T^{3} + 39717 T^{4} + 49707 T^{5} + 3717309 T^{6} + 49707 p T^{7} + 39717 p^{2} T^{8} + 1330 p^{3} T^{9} + 306 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \) |
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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
−5.49373225772977861423087889025, −4.89122208013654344516639969902, −4.83798062771261376120918357930, −4.81496306487019031816109546918, −4.72872911849213580370538934219, −4.43914753725609726934529766603, −4.03684670597464169344544427173, −3.90961813779821085817641056233, −3.88040862690190676346678927217, −3.69357638218139048366660404193, −3.63935858797945197202978142971, −3.38589884543918304546029370766, −3.29377147504321938504406966702, −3.16023219447559000108401861847, −2.93904173968376261911089112668, −2.74370124041999680862272381307, −2.44580249717101657614274819911, −2.42694756267986886009669496741, −2.28141745164990811502721704948, −1.61260937413580902929413905316, −1.38616817106571852617957574524, −1.35470255630095508169848876718, −1.12057135072876904889104095088, −1.10257065928358943307792487722, −0.958441805206467784093265928495,
0.958441805206467784093265928495, 1.10257065928358943307792487722, 1.12057135072876904889104095088, 1.35470255630095508169848876718, 1.38616817106571852617957574524, 1.61260937413580902929413905316, 2.28141745164990811502721704948, 2.42694756267986886009669496741, 2.44580249717101657614274819911, 2.74370124041999680862272381307, 2.93904173968376261911089112668, 3.16023219447559000108401861847, 3.29377147504321938504406966702, 3.38589884543918304546029370766, 3.63935858797945197202978142971, 3.69357638218139048366660404193, 3.88040862690190676346678927217, 3.90961813779821085817641056233, 4.03684670597464169344544427173, 4.43914753725609726934529766603, 4.72872911849213580370538934219, 4.81496306487019031816109546918, 4.83798062771261376120918357930, 4.89122208013654344516639969902, 5.49373225772977861423087889025
Plot not available for L-functions of degree greater than 10.
