| L(s) = 1 | − 4·5-s − 8·7-s − 16·13-s − 2·16-s + 4·17-s + 8·19-s + 8·25-s − 12·29-s + 8·31-s + 32·35-s − 4·37-s + 12·41-s + 16·49-s − 40·53-s − 8·59-s + 64·65-s + 32·67-s + 12·71-s − 20·73-s + 24·79-s + 8·80-s + 44·83-s − 16·85-s + 16·89-s + 128·91-s − 32·95-s + 8·97-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 3.02·7-s − 4.43·13-s − 1/2·16-s + 0.970·17-s + 1.83·19-s + 8/5·25-s − 2.22·29-s + 1.43·31-s + 5.40·35-s − 0.657·37-s + 1.87·41-s + 16/7·49-s − 5.49·53-s − 1.04·59-s + 7.93·65-s + 3.90·67-s + 1.42·71-s − 2.34·73-s + 2.70·79-s + 0.894·80-s + 4.82·83-s − 1.73·85-s + 1.69·89-s + 13.4·91-s − 3.28·95-s + 0.812·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 7^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 7^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.061645330\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.061645330\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{4} )^{2} \) |
| 3 | \( 1 \) |
| 7 | \( ( 1 + 4 T + 16 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | \( 1 + 16 T + 114 T^{2} + 512 T^{3} + 1890 T^{4} + 512 p T^{5} + 114 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| good | 5 | \( 1 + 4 T + 8 T^{2} + 16 T^{3} + 41 T^{4} + 128 T^{5} + 312 T^{6} + 828 T^{7} + 2176 T^{8} + 828 p T^{9} + 312 p^{2} T^{10} + 128 p^{3} T^{11} + 41 p^{4} T^{12} + 16 p^{5} T^{13} + 8 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 11 | \( 1 + 48 T^{3} + 57 T^{4} - 768 T^{5} + 1152 T^{6} - 1872 T^{7} - 34240 T^{8} - 1872 p T^{9} + 1152 p^{2} T^{10} - 768 p^{3} T^{11} + 57 p^{4} T^{12} + 48 p^{5} T^{13} + p^{8} T^{16} \) |
| 17 | \( ( 1 - 2 T + 45 T^{2} - 134 T^{3} + 944 T^{4} - 134 p T^{5} + 45 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 19 | \( 1 - 8 T + 32 T^{2} - 176 T^{3} + 51 p T^{4} - 3200 T^{5} + 10080 T^{6} - 36696 T^{7} + 117184 T^{8} - 36696 p T^{9} + 10080 p^{2} T^{10} - 3200 p^{3} T^{11} + 51 p^{5} T^{12} - 176 p^{5} T^{13} + 32 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) |
| 23 | \( 1 - 38 T^{2} + 1333 T^{4} - 32994 T^{6} + 926696 T^{8} - 32994 p^{2} T^{10} + 1333 p^{4} T^{12} - 38 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( ( 1 + 6 T + 71 T^{2} + 298 T^{3} + 2534 T^{4} + 298 p T^{5} + 71 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( 1 - 8 T + 32 T^{2} - 216 T^{3} + 2372 T^{4} - 14392 T^{5} + 62560 T^{6} - 474792 T^{7} + 3604870 T^{8} - 474792 p T^{9} + 62560 p^{2} T^{10} - 14392 p^{3} T^{11} + 2372 p^{4} T^{12} - 216 p^{5} T^{13} + 32 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) |
| 37 | \( 1 + 4 T + 8 T^{2} + 112 T^{3} + 3465 T^{4} + 11872 T^{5} + 26040 T^{6} + 394524 T^{7} + 5896384 T^{8} + 394524 p T^{9} + 26040 p^{2} T^{10} + 11872 p^{3} T^{11} + 3465 p^{4} T^{12} + 112 p^{5} T^{13} + 8 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 41 | \( 1 - 12 T + 72 T^{2} - 436 T^{3} + 2176 T^{4} - 11692 T^{5} + 78680 T^{6} - 488660 T^{7} + 3032766 T^{8} - 488660 p T^{9} + 78680 p^{2} T^{10} - 11692 p^{3} T^{11} + 2176 p^{4} T^{12} - 436 p^{5} T^{13} + 72 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) |
| 43 | \( 1 - 6 p T^{2} + 31169 T^{4} - 2332954 T^{6} + 119560116 T^{8} - 2332954 p^{2} T^{10} + 31169 p^{4} T^{12} - 6 p^{7} T^{14} + p^{8} T^{16} \) |
| 47 | \( ( 1 + 3682 T^{4} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + 20 T + 278 T^{2} + 2892 T^{3} + 23978 T^{4} + 2892 p T^{5} + 278 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 59 | \( 1 + 8 T + 32 T^{2} - 248 T^{3} + 2076 T^{4} - 7864 T^{5} - 98592 T^{6} - 1411448 T^{7} + 8207270 T^{8} - 1411448 p T^{9} - 98592 p^{2} T^{10} - 7864 p^{3} T^{11} + 2076 p^{4} T^{12} - 248 p^{5} T^{13} + 32 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) |
| 61 | \( 1 + 14 T^{2} + 7685 T^{4} + 105546 T^{6} + 30490872 T^{8} + 105546 p^{2} T^{10} + 7685 p^{4} T^{12} + 14 p^{6} T^{14} + p^{8} T^{16} \) |
| 67 | \( 1 - 32 T + 512 T^{2} - 5600 T^{3} + 50684 T^{4} - 415584 T^{5} + 3028480 T^{6} - 18496928 T^{7} + 122079910 T^{8} - 18496928 p T^{9} + 3028480 p^{2} T^{10} - 415584 p^{3} T^{11} + 50684 p^{4} T^{12} - 5600 p^{5} T^{13} + 512 p^{6} T^{14} - 32 p^{7} T^{15} + p^{8} T^{16} \) |
| 71 | \( 1 - 12 T + 72 T^{2} - 908 T^{3} + 2064 T^{4} + 51444 T^{5} - 353704 T^{6} + 5033588 T^{7} - 69761186 T^{8} + 5033588 p T^{9} - 353704 p^{2} T^{10} + 51444 p^{3} T^{11} + 2064 p^{4} T^{12} - 908 p^{5} T^{13} + 72 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) |
| 73 | \( 1 + 20 T + 200 T^{2} + 576 T^{3} - 9079 T^{4} - 107216 T^{5} - 162632 T^{6} + 7946652 T^{7} + 106585744 T^{8} + 7946652 p T^{9} - 162632 p^{2} T^{10} - 107216 p^{3} T^{11} - 9079 p^{4} T^{12} + 576 p^{5} T^{13} + 200 p^{6} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} \) |
| 79 | \( ( 1 - 12 T + 246 T^{2} - 1836 T^{3} + 318 p T^{4} - 1836 p T^{5} + 246 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 - 44 T + 968 T^{2} - 16060 T^{3} + 235728 T^{4} - 3037100 T^{5} + 34409496 T^{6} - 357330556 T^{7} + 3416693630 T^{8} - 357330556 p T^{9} + 34409496 p^{2} T^{10} - 3037100 p^{3} T^{11} + 235728 p^{4} T^{12} - 16060 p^{5} T^{13} + 968 p^{6} T^{14} - 44 p^{7} T^{15} + p^{8} T^{16} \) |
| 89 | \( 1 - 16 T + 128 T^{2} - 1328 T^{3} + 23068 T^{4} - 260752 T^{5} + 2101120 T^{6} - 20928816 T^{7} + 206877318 T^{8} - 20928816 p T^{9} + 2101120 p^{2} T^{10} - 260752 p^{3} T^{11} + 23068 p^{4} T^{12} - 1328 p^{5} T^{13} + 128 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) |
| 97 | \( 1 - 8 T + 32 T^{2} - 1032 T^{3} - 988 T^{4} + 101096 T^{5} - 244640 T^{6} + 6736104 T^{7} - 156517562 T^{8} + 6736104 p T^{9} - 244640 p^{2} T^{10} + 101096 p^{3} T^{11} - 988 p^{4} T^{12} - 1032 p^{5} T^{13} + 32 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.88555367365399986490565600418, −3.75913795397159205496959590867, −3.68589831173953989189671296246, −3.54838909172198946798742487282, −3.41563873544095110495727126411, −3.31054835219336672642433181524, −3.25509154993867560627974788831, −3.17391548181708714875860211826, −2.93535781916019026330856780948, −2.82920247753287371136783692925, −2.63026418181702397865238868743, −2.57598628236058992767040696070, −2.56111020932941331906763192037, −2.52583903052832469228487886699, −2.02318067733184076198186367393, −1.87726431410801738908441359651, −1.87265420368278072195789725046, −1.87168502620769290178333505402, −1.32542852485837861337782298768, −1.30118732776540395133974524502, −0.980660032658695935067248992698, −0.46864015806843788458586799419, −0.45868108599408485358168833652, −0.42157615334557521590622431412, −0.31718871934922188188813968764,
0.31718871934922188188813968764, 0.42157615334557521590622431412, 0.45868108599408485358168833652, 0.46864015806843788458586799419, 0.980660032658695935067248992698, 1.30118732776540395133974524502, 1.32542852485837861337782298768, 1.87168502620769290178333505402, 1.87265420368278072195789725046, 1.87726431410801738908441359651, 2.02318067733184076198186367393, 2.52583903052832469228487886699, 2.56111020932941331906763192037, 2.57598628236058992767040696070, 2.63026418181702397865238868743, 2.82920247753287371136783692925, 2.93535781916019026330856780948, 3.17391548181708714875860211826, 3.25509154993867560627974788831, 3.31054835219336672642433181524, 3.41563873544095110495727126411, 3.54838909172198946798742487282, 3.68589831173953989189671296246, 3.75913795397159205496959590867, 3.88555367365399986490565600418
Plot not available for L-functions of degree greater than 10.
