| L(s) = 1 | − 3·2-s + 3·3-s + 3·4-s + 2·5-s − 9·6-s + 12·7-s + 2·8-s + 3·9-s − 6·10-s + 6·11-s + 9·12-s + 3·13-s − 36·14-s + 6·15-s − 9·16-s − 2·17-s − 9·18-s + 3·19-s + 6·20-s + 36·21-s − 18·22-s + 10·23-s + 6·24-s + 9·25-s − 9·26-s − 2·27-s + 36·28-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 1.73·3-s + 3/2·4-s + 0.894·5-s − 3.67·6-s + 4.53·7-s + 0.707·8-s + 9-s − 1.89·10-s + 1.80·11-s + 2.59·12-s + 0.832·13-s − 9.62·14-s + 1.54·15-s − 9/4·16-s − 0.485·17-s − 2.12·18-s + 0.688·19-s + 1.34·20-s + 7.85·21-s − 3.83·22-s + 2.08·23-s + 1.22·24-s + 9/5·25-s − 1.76·26-s − 0.384·27-s + 6.80·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 11^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 11^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.895290416\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.895290416\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T + T^{2} )^{3} \) |
| 3 | \( ( 1 - T + T^{2} )^{3} \) |
| 11 | \( ( 1 - T )^{6} \) |
| 19 | \( 1 - 3 T - 6 T^{2} + 43 T^{3} - 6 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| good | 5 | \( 1 - 2 T - p T^{2} + 6 T^{3} + 12 T^{4} + 22 T^{5} - 111 T^{6} + 22 p T^{7} + 12 p^{2} T^{8} + 6 p^{3} T^{9} - p^{5} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \) |
| 7 | \( ( 1 - 6 T + 26 T^{2} - 76 T^{3} + 26 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 - 3 T + 6 T^{2} + 3 p T^{3} - 102 T^{4} - 345 T^{5} + 3332 T^{6} - 345 p T^{7} - 102 p^{2} T^{8} + 3 p^{4} T^{9} + 6 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 17 | \( 1 + 2 T - 16 T^{2} - 184 T^{3} - 164 T^{4} + 1458 T^{5} + 14414 T^{6} + 1458 p T^{7} - 164 p^{2} T^{8} - 184 p^{3} T^{9} - 16 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \) |
| 23 | \( 1 - 10 T + 14 T^{2} + 68 T^{3} + 778 T^{4} - 5634 T^{5} + 16622 T^{6} - 5634 p T^{7} + 778 p^{2} T^{8} + 68 p^{3} T^{9} + 14 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \) |
| 29 | \( 1 - 3 T - 47 T^{2} - 38 T^{3} + 1317 T^{4} + 3581 T^{5} - 45510 T^{6} + 3581 p T^{7} + 1317 p^{2} T^{8} - 38 p^{3} T^{9} - 47 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 31 | \( ( 1 + 10 T + 119 T^{2} + 636 T^{3} + 119 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( ( 1 + 4 T + p T^{2} )^{6} \) |
| 41 | \( 1 + 8 T - 11 T^{2} - 456 T^{3} - 1674 T^{4} + 3848 T^{5} + 78417 T^{6} + 3848 p T^{7} - 1674 p^{2} T^{8} - 456 p^{3} T^{9} - 11 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \) |
| 43 | \( 1 + 7 T - 41 T^{2} - 620 T^{3} - 145 T^{4} + 15013 T^{5} + 110738 T^{6} + 15013 p T^{7} - 145 p^{2} T^{8} - 620 p^{3} T^{9} - 41 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \) |
| 47 | \( 1 - 7 T - p T^{2} + 30 T^{3} + 2247 T^{4} + 14009 T^{5} - 210546 T^{6} + 14009 p T^{7} + 2247 p^{2} T^{8} + 30 p^{3} T^{9} - p^{5} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \) |
| 53 | \( 1 + 26 T + 309 T^{2} + 3038 T^{3} + 30686 T^{4} + 255066 T^{5} + 1835857 T^{6} + 255066 p T^{7} + 30686 p^{2} T^{8} + 3038 p^{3} T^{9} + 309 p^{4} T^{10} + 26 p^{5} T^{11} + p^{6} T^{12} \) |
| 59 | \( 1 + 2 T - 157 T^{2} - 2 p T^{3} + 15982 T^{4} + 4386 T^{5} - 1084105 T^{6} + 4386 p T^{7} + 15982 p^{2} T^{8} - 2 p^{4} T^{9} - 157 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \) |
| 61 | \( 1 + 3 T - 2 T^{2} - 851 T^{3} - 1876 T^{4} + 11999 T^{5} + 608456 T^{6} + 11999 p T^{7} - 1876 p^{2} T^{8} - 851 p^{3} T^{9} - 2 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 67 | \( 1 + 6 T - 111 T^{2} - 294 T^{3} + 8232 T^{4} - 4818 T^{5} - 662137 T^{6} - 4818 p T^{7} + 8232 p^{2} T^{8} - 294 p^{3} T^{9} - 111 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 71 | \( 1 - 7 T - 163 T^{2} + 458 T^{3} + 23095 T^{4} - 30411 T^{5} - 1779370 T^{6} - 30411 p T^{7} + 23095 p^{2} T^{8} + 458 p^{3} T^{9} - 163 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \) |
| 73 | \( 1 + 6 T - 59 T^{2} + 562 T^{3} + 2066 T^{4} - 50866 T^{5} - 72619 T^{6} - 50866 p T^{7} + 2066 p^{2} T^{8} + 562 p^{3} T^{9} - 59 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 79 | \( 1 - 6 T - 101 T^{2} + 386 T^{3} + 4322 T^{4} + 9682 T^{5} - 356533 T^{6} + 9682 p T^{7} + 4322 p^{2} T^{8} + 386 p^{3} T^{9} - 101 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 83 | \( ( 1 + 19 T + 353 T^{2} + 3326 T^{3} + 353 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 89 | \( 1 - 37 T + 678 T^{2} - 9571 T^{3} + 120056 T^{4} - 1268769 T^{5} + 12038284 T^{6} - 1268769 p T^{7} + 120056 p^{2} T^{8} - 9571 p^{3} T^{9} + 678 p^{4} T^{10} - 37 p^{5} T^{11} + p^{6} T^{12} \) |
| 97 | \( 1 + T - 129 T^{2} + 800 T^{3} + 4625 T^{4} - 59361 T^{5} + 288862 T^{6} - 59361 p T^{7} + 4625 p^{2} T^{8} + 800 p^{3} T^{9} - 129 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \) |
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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
−5.13133271252072893674056400176, −5.01349495829686994755413251962, −4.75070612325496286856383644143, −4.73377002026592167999445455014, −4.56702763736243982602026955342, −4.44499346451817828377566800751, −4.02640903312572756686300734110, −3.90413429356284829533504128080, −3.87563390138396654404110419693, −3.52641340079845831945129214393, −3.27431575724861398457916195238, −3.27069710822891362442496794362, −3.16018714334800215031856824368, −2.98950970044281064632847971613, −2.52816104740058299561546715425, −2.38198106122704904174628813912, −1.86458780566143654213150004210, −1.82586245945233310526662749245, −1.76098107566736558313097400787, −1.69273634176523288362704593174, −1.57646064633531466723583618273, −1.43421892080224852458218805044, −1.09728472047749531136290061185, −0.893708026341500811278208609820, −0.28421253234324838221673942225,
0.28421253234324838221673942225, 0.893708026341500811278208609820, 1.09728472047749531136290061185, 1.43421892080224852458218805044, 1.57646064633531466723583618273, 1.69273634176523288362704593174, 1.76098107566736558313097400787, 1.82586245945233310526662749245, 1.86458780566143654213150004210, 2.38198106122704904174628813912, 2.52816104740058299561546715425, 2.98950970044281064632847971613, 3.16018714334800215031856824368, 3.27069710822891362442496794362, 3.27431575724861398457916195238, 3.52641340079845831945129214393, 3.87563390138396654404110419693, 3.90413429356284829533504128080, 4.02640903312572756686300734110, 4.44499346451817828377566800751, 4.56702763736243982602026955342, 4.73377002026592167999445455014, 4.75070612325496286856383644143, 5.01349495829686994755413251962, 5.13133271252072893674056400176
Plot not available for L-functions of degree greater than 10.
