| L(s) = 1 | − 2·3-s + 5·4-s + 2·5-s − 4·7-s + 2·9-s − 10·12-s − 4·15-s + 10·16-s − 8·17-s + 14·19-s + 10·20-s + 8·21-s + 25-s + 4·27-s − 20·28-s − 6·29-s − 8·35-s + 10·36-s − 6·37-s + 4·45-s − 8·47-s − 20·48-s + 8·49-s + 16·51-s − 14·53-s − 28·57-s + 24·59-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 5/2·4-s + 0.894·5-s − 1.51·7-s + 2/3·9-s − 2.88·12-s − 1.03·15-s + 5/2·16-s − 1.94·17-s + 3.21·19-s + 2.23·20-s + 1.74·21-s + 1/5·25-s + 0.769·27-s − 3.77·28-s − 1.11·29-s − 1.35·35-s + 5/3·36-s − 0.986·37-s + 0.596·45-s − 1.16·47-s − 2.88·48-s + 8/7·49-s + 2.24·51-s − 1.92·53-s − 3.70·57-s + 3.12·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 37^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 37^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.954341622\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.954341622\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 2 T + 3 T^{2} - 12 T^{3} + 3 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| good | 2 | \( 1 - 5 T^{2} + 15 T^{4} - 35 T^{6} + 15 p^{2} T^{8} - 5 p^{4} T^{10} + p^{6} T^{12} \) |
| 3 | \( 1 + 2 T + 2 T^{2} - 4 T^{3} - 5 T^{4} + 22 T^{5} + 62 T^{6} + 22 p T^{7} - 5 p^{2} T^{8} - 4 p^{3} T^{9} + 2 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \) |
| 7 | \( 1 + 4 T + 8 T^{2} + 30 T^{3} + 95 T^{4} + 146 T^{5} + 274 T^{6} + 146 p T^{7} + 95 p^{2} T^{8} + 30 p^{3} T^{9} + 8 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \) |
| 11 | \( 1 + 2 T^{2} + 279 T^{4} + 716 T^{6} + 279 p^{2} T^{8} + 2 p^{4} T^{10} + p^{6} T^{12} \) |
| 13 | \( 1 - 14 T^{2} + 391 T^{4} - 4352 T^{6} + 391 p^{2} T^{8} - 14 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( ( 1 + 4 T + 43 T^{2} + 6 p T^{3} + 43 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 19 | \( 1 - 14 T + 98 T^{2} - 624 T^{3} + 3995 T^{4} - 20446 T^{5} + 89422 T^{6} - 20446 p T^{7} + 3995 p^{2} T^{8} - 624 p^{3} T^{9} + 98 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \) |
| 23 | \( 1 - 70 T^{2} + 1887 T^{4} - 37060 T^{6} + 1887 p^{2} T^{8} - 70 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( 1 + 6 T + 18 T^{2} + 14 T^{3} + 183 T^{4} + 5812 T^{5} + 31676 T^{6} + 5812 p T^{7} + 183 p^{2} T^{8} + 14 p^{3} T^{9} + 18 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 31 | \( 1 + 134 T^{3} - 817 T^{4} - 5762 T^{5} + 8978 T^{6} - 5762 p T^{7} - 817 p^{2} T^{8} + 134 p^{3} T^{9} + p^{6} T^{12} \) |
| 41 | \( 1 + 18 T^{2} + 1775 T^{4} + 40492 T^{6} + 1775 p^{2} T^{8} + 18 p^{4} T^{10} + p^{6} T^{12} \) |
| 43 | \( 1 - 98 T^{2} + 7543 T^{4} - 373484 T^{6} + 7543 p^{2} T^{8} - 98 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 + 8 T + 32 T^{2} + 38 T^{3} + 1155 T^{4} + 27058 T^{5} + 180226 T^{6} + 27058 p T^{7} + 1155 p^{2} T^{8} + 38 p^{3} T^{9} + 32 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \) |
| 53 | \( 1 + 14 T + 98 T^{2} + 1086 T^{3} + 14615 T^{4} + 101644 T^{5} + 580444 T^{6} + 101644 p T^{7} + 14615 p^{2} T^{8} + 1086 p^{3} T^{9} + 98 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \) |
| 59 | \( 1 - 24 T + 288 T^{2} - 2798 T^{3} + 23415 T^{4} - 170230 T^{5} + 1256402 T^{6} - 170230 p T^{7} + 23415 p^{2} T^{8} - 2798 p^{3} T^{9} + 288 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \) |
| 61 | \( 1 - 26 T + 338 T^{2} - 4002 T^{3} + 46455 T^{4} - 422524 T^{5} + 3291836 T^{6} - 422524 p T^{7} + 46455 p^{2} T^{8} - 4002 p^{3} T^{9} + 338 p^{4} T^{10} - 26 p^{5} T^{11} + p^{6} T^{12} \) |
| 67 | \( 1 - 22 T + 242 T^{2} - 24 p T^{3} + 15111 T^{4} - 205946 T^{5} + 2166782 T^{6} - 205946 p T^{7} + 15111 p^{2} T^{8} - 24 p^{4} T^{9} + 242 p^{4} T^{10} - 22 p^{5} T^{11} + p^{6} T^{12} \) |
| 71 | \( ( 1 - 2 T + 161 T^{2} - 100 T^{3} + 161 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 + 38 T + 722 T^{2} + 9790 T^{3} + 110271 T^{4} + 1090188 T^{5} + 9733532 T^{6} + 1090188 p T^{7} + 110271 p^{2} T^{8} + 9790 p^{3} T^{9} + 722 p^{4} T^{10} + 38 p^{5} T^{11} + p^{6} T^{12} \) |
| 79 | \( 1 + 24 T + 288 T^{2} + 3586 T^{3} + 48515 T^{4} + 485606 T^{5} + 4111922 T^{6} + 485606 p T^{7} + 48515 p^{2} T^{8} + 3586 p^{3} T^{9} + 288 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \) |
| 83 | \( 1 - 18 T + 162 T^{2} - 2156 T^{3} + 33111 T^{4} - 284650 T^{5} + 2083886 T^{6} - 284650 p T^{7} + 33111 p^{2} T^{8} - 2156 p^{3} T^{9} + 162 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \) |
| 89 | \( 1 + 6 T + 18 T^{2} + 494 T^{3} + 4623 T^{4} - 132 p T^{5} - 4 p^{2} T^{6} - 132 p^{2} T^{7} + 4623 p^{2} T^{8} + 494 p^{3} T^{9} + 18 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 97 | \( ( 1 + 2 T + 283 T^{2} + 392 T^{3} + 283 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
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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
−6.89554731764850968695159465336, −6.84864841944309190451424492434, −6.72783829437329737250072071492, −6.48083200342084485608789702288, −6.37168238125603351793417893014, −5.89267320017335200137556511034, −5.78912548711155256315166067240, −5.71374780943575609470727471395, −5.57682970148646860977499302710, −5.47302170901219952041803599685, −4.95813127193799505888560496932, −4.82189375207204631197551845640, −4.66906855738295981465207300110, −4.39431921498008810296840134600, −3.91584371697693271680763101391, −3.58550168408044011024846684626, −3.48787935946983672189902215610, −3.13093798439865244708666706878, −3.10600102340992821212737598535, −2.48735242295633432778536816304, −2.45540837306857380680161182328, −2.17047087426007556117523861101, −1.79950073499897033955919851265, −1.42020209741084680288268981142, −0.74612836214952433856787255915,
0.74612836214952433856787255915, 1.42020209741084680288268981142, 1.79950073499897033955919851265, 2.17047087426007556117523861101, 2.45540837306857380680161182328, 2.48735242295633432778536816304, 3.10600102340992821212737598535, 3.13093798439865244708666706878, 3.48787935946983672189902215610, 3.58550168408044011024846684626, 3.91584371697693271680763101391, 4.39431921498008810296840134600, 4.66906855738295981465207300110, 4.82189375207204631197551845640, 4.95813127193799505888560496932, 5.47302170901219952041803599685, 5.57682970148646860977499302710, 5.71374780943575609470727471395, 5.78912548711155256315166067240, 5.89267320017335200137556511034, 6.37168238125603351793417893014, 6.48083200342084485608789702288, 6.72783829437329737250072071492, 6.84864841944309190451424492434, 6.89554731764850968695159465336
Plot not available for L-functions of degree greater than 10.
