| L(s) = 1 | − 6·3-s + 6·7-s + 14·9-s + 12·11-s − 18·17-s − 36·21-s − 6·23-s − 12·27-s − 72·33-s + 18·37-s + 12·41-s + 18·43-s + 34·49-s + 108·51-s + 24·59-s − 4·61-s + 84·63-s − 18·67-s + 36·69-s − 36·71-s + 48·73-s + 72·77-s + 16·79-s − 5·81-s + 72·83-s + 24·89-s − 6·97-s + ⋯ |
| L(s) = 1 | − 3.46·3-s + 2.26·7-s + 14/3·9-s + 3.61·11-s − 4.36·17-s − 7.85·21-s − 1.25·23-s − 2.30·27-s − 12.5·33-s + 2.95·37-s + 1.87·41-s + 2.74·43-s + 34/7·49-s + 15.1·51-s + 3.12·59-s − 0.512·61-s + 10.5·63-s − 2.19·67-s + 4.33·69-s − 4.27·71-s + 5.61·73-s + 8.20·77-s + 1.80·79-s − 5/9·81-s + 7.90·83-s + 2.54·89-s − 0.609·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.170489058\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.170489058\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + 16 T^{2} + 96 T^{3} + 30 T^{4} + 96 p T^{5} + 16 p^{2} T^{6} + p^{4} T^{8} \) |
| good | 3 | \( 1 + 2 p T + 22 T^{2} + 20 p T^{3} + 43 p T^{4} + 28 p^{2} T^{5} + 466 T^{6} + 94 p^{2} T^{7} + 1516 T^{8} + 94 p^{3} T^{9} + 466 p^{2} T^{10} + 28 p^{5} T^{11} + 43 p^{5} T^{12} + 20 p^{6} T^{13} + 22 p^{6} T^{14} + 2 p^{8} T^{15} + p^{8} T^{16} \) |
| 7 | \( 1 - 6 T + 2 T^{2} + 36 T^{3} + 61 T^{4} - 60 p T^{5} - 58 T^{6} + 66 T^{7} + 5476 T^{8} + 66 p T^{9} - 58 p^{2} T^{10} - 60 p^{4} T^{11} + 61 p^{4} T^{12} + 36 p^{5} T^{13} + 2 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 11 | \( ( 1 - 3 T + 14 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{4} \) |
| 17 | \( 1 + 18 T + 194 T^{2} + 1548 T^{3} + 10261 T^{4} + 59616 T^{5} + 310430 T^{6} + 1467810 T^{7} + 6325732 T^{8} + 1467810 p T^{9} + 310430 p^{2} T^{10} + 59616 p^{3} T^{11} + 10261 p^{4} T^{12} + 1548 p^{5} T^{13} + 194 p^{6} T^{14} + 18 p^{7} T^{15} + p^{8} T^{16} \) |
| 19 | \( 1 + 46 T^{2} + 1057 T^{4} + 15502 T^{6} + 227284 T^{8} + 15502 p^{2} T^{10} + 1057 p^{4} T^{12} + 46 p^{6} T^{14} + p^{8} T^{16} \) |
| 23 | \( 1 + 6 T + 74 T^{2} + 372 T^{3} + 2989 T^{4} + 18252 T^{5} + 96926 T^{6} + 568398 T^{7} + 2281540 T^{8} + 568398 p T^{9} + 96926 p^{2} T^{10} + 18252 p^{3} T^{11} + 2989 p^{4} T^{12} + 372 p^{5} T^{13} + 74 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 29 | \( 1 - 74 T^{2} + 192 T^{3} + 2905 T^{4} - 9888 T^{5} - 56570 T^{6} + 165216 T^{7} + 1030180 T^{8} + 165216 p T^{9} - 56570 p^{2} T^{10} - 9888 p^{3} T^{11} + 2905 p^{4} T^{12} + 192 p^{5} T^{13} - 74 p^{6} T^{14} + p^{8} T^{16} \) |
| 31 | \( ( 1 - 28 T^{2} + 1926 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( 1 - 18 T + 110 T^{2} - 180 T^{3} - 143 T^{4} + 3240 T^{5} - 24622 T^{6} - 778122 T^{7} + 9708988 T^{8} - 778122 p T^{9} - 24622 p^{2} T^{10} + 3240 p^{3} T^{11} - 143 p^{4} T^{12} - 180 p^{5} T^{13} + 110 p^{6} T^{14} - 18 p^{7} T^{15} + p^{8} T^{16} \) |
| 41 | \( ( 1 - 6 T + 61 T^{2} - 294 T^{3} + 1212 T^{4} - 294 p T^{5} + 61 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 - 18 T + 218 T^{2} - 1980 T^{3} + 13117 T^{4} - 50652 T^{5} - 24226 T^{6} + 2431782 T^{7} - 22389836 T^{8} + 2431782 p T^{9} - 24226 p^{2} T^{10} - 50652 p^{3} T^{11} + 13117 p^{4} T^{12} - 1980 p^{5} T^{13} + 218 p^{6} T^{14} - 18 p^{7} T^{15} + p^{8} T^{16} \) |
| 47 | \( ( 1 + 82 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( 1 + 32 T^{2} + 2812 T^{4} - 295456 T^{6} - 9399194 T^{8} - 295456 p^{2} T^{10} + 2812 p^{4} T^{12} + 32 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( 1 - 24 T + 410 T^{2} - 5232 T^{3} + 55429 T^{4} - 515160 T^{5} + 4379474 T^{6} - 35226912 T^{7} + 274579084 T^{8} - 35226912 p T^{9} + 4379474 p^{2} T^{10} - 515160 p^{3} T^{11} + 55429 p^{4} T^{12} - 5232 p^{5} T^{13} + 410 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \) |
| 61 | \( 1 + 4 T - 126 T^{2} + 392 T^{3} + 10901 T^{4} - 53424 T^{5} - 222326 T^{6} + 2281108 T^{7} - 2679156 T^{8} + 2281108 p T^{9} - 222326 p^{2} T^{10} - 53424 p^{3} T^{11} + 10901 p^{4} T^{12} + 392 p^{5} T^{13} - 126 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 67 | \( 1 + 18 T + 62 T^{2} - 900 T^{3} - 9071 T^{4} - 54324 T^{5} - 268150 T^{6} + 4296762 T^{7} + 76577740 T^{8} + 4296762 p T^{9} - 268150 p^{2} T^{10} - 54324 p^{3} T^{11} - 9071 p^{4} T^{12} - 900 p^{5} T^{13} + 62 p^{6} T^{14} + 18 p^{7} T^{15} + p^{8} T^{16} \) |
| 71 | \( 1 + 36 T + 698 T^{2} + 9576 T^{3} + 99037 T^{4} + 800280 T^{5} + 5013074 T^{6} + 24891156 T^{7} + 146900620 T^{8} + 24891156 p T^{9} + 5013074 p^{2} T^{10} + 800280 p^{3} T^{11} + 99037 p^{4} T^{12} + 9576 p^{5} T^{13} + 698 p^{6} T^{14} + 36 p^{7} T^{15} + p^{8} T^{16} \) |
| 73 | \( ( 1 - 24 T + 448 T^{2} - 72 p T^{3} + 53166 T^{4} - 72 p^{2} T^{5} + 448 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 8 T + 136 T^{2} - 392 T^{3} + 8638 T^{4} - 392 p T^{5} + 136 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 - 36 T + 740 T^{2} - 10404 T^{3} + 109638 T^{4} - 10404 p T^{5} + 740 p^{2} T^{6} - 36 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 89 | \( 1 - 24 T + 530 T^{2} - 8112 T^{3} + 116461 T^{4} - 1452168 T^{5} + 16850810 T^{6} - 177993840 T^{7} + 1745476972 T^{8} - 177993840 p T^{9} + 16850810 p^{2} T^{10} - 1452168 p^{3} T^{11} + 116461 p^{4} T^{12} - 8112 p^{5} T^{13} + 530 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \) |
| 97 | \( 1 + 6 T - 166 T^{2} + 1380 T^{3} + 253 p T^{4} - 226320 T^{5} + 470246 T^{6} + 18763110 T^{7} - 126293804 T^{8} + 18763110 p T^{9} + 470246 p^{2} T^{10} - 226320 p^{3} T^{11} + 253 p^{5} T^{12} + 1380 p^{5} T^{13} - 166 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.22097461475760230223807816297, −4.10219181807112605275174924556, −4.03870642135082724186806745200, −3.91100051984610412793792430019, −3.65486414735108444743906210927, −3.55739724683567893273355330441, −3.48303080143623825665877018167, −3.44497050058262409016696462640, −3.15934029235745371670703384935, −2.68467279022884841184847076052, −2.50801347458485083862423814699, −2.48302270644912475930481957924, −2.42766787173044470494694502104, −2.29443525023836464638073113049, −2.09281571528914665245582315250, −2.08694892282751372120343049559, −1.90000672681338178520342796832, −1.73351797843399501386692177036, −1.34407879008367188942025133066, −1.19833562203825573771158789316, −0.828515229482971186313960288763, −0.824475140068356627057962390120, −0.798565267865760262825232877817, −0.72137056943442403303494806816, −0.16402553179957364535387435571,
0.16402553179957364535387435571, 0.72137056943442403303494806816, 0.798565267865760262825232877817, 0.824475140068356627057962390120, 0.828515229482971186313960288763, 1.19833562203825573771158789316, 1.34407879008367188942025133066, 1.73351797843399501386692177036, 1.90000672681338178520342796832, 2.08694892282751372120343049559, 2.09281571528914665245582315250, 2.29443525023836464638073113049, 2.42766787173044470494694502104, 2.48302270644912475930481957924, 2.50801347458485083862423814699, 2.68467279022884841184847076052, 3.15934029235745371670703384935, 3.44497050058262409016696462640, 3.48303080143623825665877018167, 3.55739724683567893273355330441, 3.65486414735108444743906210927, 3.91100051984610412793792430019, 4.03870642135082724186806745200, 4.10219181807112605275174924556, 4.22097461475760230223807816297
Plot not available for L-functions of degree greater than 10.
