| L(s) = 1 | + 2·3-s − 3·4-s − 6·7-s + 6·9-s − 6·12-s + 5·16-s − 2·17-s + 12·19-s − 12·21-s − 10·23-s − 4·25-s + 12·27-s + 18·28-s − 8·29-s − 6·32-s − 18·36-s + 6·37-s + 12·41-s − 2·43-s + 10·48-s + 10·49-s − 4·51-s − 24·53-s + 24·57-s − 12·59-s − 28·61-s − 36·63-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 3/2·4-s − 2.26·7-s + 2·9-s − 1.73·12-s + 5/4·16-s − 0.485·17-s + 2.75·19-s − 2.61·21-s − 2.08·23-s − 4/5·25-s + 2.30·27-s + 3.40·28-s − 1.48·29-s − 1.06·32-s − 3·36-s + 0.986·37-s + 1.87·41-s − 0.304·43-s + 1.44·48-s + 10/7·49-s − 0.560·51-s − 3.29·53-s + 3.17·57-s − 1.56·59-s − 3.58·61-s − 4.53·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3775126709\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3775126709\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( ( 1 + T^{2} )^{4} \) |
| 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 | 2 | \( ( 1 - p T + 5 T^{2} - p^{3} T^{3} + 13 T^{4} - p^{4} T^{5} + 5 p^{2} T^{6} - p^{4} T^{7} + p^{4} T^{8} )( 1 + p T + p T^{2} + p T^{3} + T^{4} + p^{2} T^{5} + p^{3} T^{6} + p^{4} T^{7} + p^{4} T^{8} ) \) |
| 3 | \( 1 - 2 T - 2 T^{2} + 4 T^{3} + T^{4} + 4 T^{5} + 10 T^{6} - 26 T^{7} - 20 T^{8} - 26 p T^{9} + 10 p^{2} T^{10} + 4 p^{3} T^{11} + p^{4} T^{12} + 4 p^{5} T^{13} - 2 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 7 | \( 1 + 6 T + 26 T^{2} + 12 p T^{3} + 229 T^{4} + 60 p T^{5} + 206 T^{6} - 954 T^{7} - 4268 T^{8} - 954 p T^{9} + 206 p^{2} T^{10} + 60 p^{4} T^{11} + 229 p^{4} T^{12} + 12 p^{6} T^{13} + 26 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 11 | \( 1 + 14 T^{2} + 97 T^{4} - 182 p T^{6} - 236 p^{2} T^{8} - 182 p^{3} T^{10} + 97 p^{4} T^{12} + 14 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( ( 1 - 14 T + 50 T^{2} + 220 T^{3} - 2129 T^{4} + 220 p T^{5} + 50 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )( 1 + 16 T + 128 T^{2} + 688 T^{3} + 3022 T^{4} + 688 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} ) \) |
| 19 | \( ( 1 - 6 T + 37 T^{2} - 150 T^{3} + 492 T^{4} - 150 p T^{5} + 37 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 23 | \( 1 + 10 T + 2 T^{2} - 108 T^{3} + 1013 T^{4} + 2804 T^{5} - 34298 T^{6} - 4266 p T^{7} + 8492 p T^{8} - 4266 p^{2} T^{9} - 34298 p^{2} T^{10} + 2804 p^{3} T^{11} + 1013 p^{4} T^{12} - 108 p^{5} T^{13} + 2 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) |
| 29 | \( 1 + 8 T - 34 T^{2} - 528 T^{3} + 353 T^{4} + 19264 T^{5} + 50686 T^{6} - 248280 T^{7} - 2118188 T^{8} - 248280 p T^{9} + 50686 p^{2} T^{10} + 19264 p^{3} T^{11} + 353 p^{4} T^{12} - 528 p^{5} T^{13} - 34 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) |
| 31 | \( ( 1 - 92 T^{2} + 3846 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( 1 - 6 T + 110 T^{2} - 588 T^{3} + 6289 T^{4} - 36600 T^{5} + 249746 T^{6} - 1600110 T^{7} + 8587516 T^{8} - 1600110 p T^{9} + 249746 p^{2} T^{10} - 36600 p^{3} T^{11} + 6289 p^{4} T^{12} - 588 p^{5} T^{13} + 110 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 41 | \( ( 1 - 6 T + 93 T^{2} - 486 T^{3} + 5372 T^{4} - 486 p T^{5} + 93 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 + 2 T - 150 T^{2} - 188 T^{3} + 13661 T^{4} + 10620 T^{5} - 862418 T^{6} - 185374 T^{7} + 42096180 T^{8} - 185374 p T^{9} - 862418 p^{2} T^{10} + 10620 p^{3} T^{11} + 13661 p^{4} T^{12} - 188 p^{5} T^{13} - 150 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 47 | \( 1 - 168 T^{2} + 17500 T^{4} - 1260696 T^{6} + 68079174 T^{8} - 1260696 p^{2} T^{10} + 17500 p^{4} T^{12} - 168 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( ( 1 + 12 T + 248 T^{2} + 36 p T^{3} + 20622 T^{4} + 36 p^{2} T^{5} + 248 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 59 | \( 1 + 12 T + 266 T^{2} + 2616 T^{3} + 37117 T^{4} + 331992 T^{5} + 3532226 T^{6} + 27597948 T^{7} + 240376924 T^{8} + 27597948 p T^{9} + 3532226 p^{2} T^{10} + 331992 p^{3} T^{11} + 37117 p^{4} T^{12} + 2616 p^{5} T^{13} + 266 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) |
| 61 | \( 1 + 28 T + 282 T^{2} + 1880 T^{3} + 26477 T^{4} + 344016 T^{5} + 2759698 T^{6} + 21027916 T^{7} + 175399068 T^{8} + 21027916 p T^{9} + 2759698 p^{2} T^{10} + 344016 p^{3} T^{11} + 26477 p^{4} T^{12} + 1880 p^{5} T^{13} + 282 p^{6} T^{14} + 28 p^{7} T^{15} + p^{8} T^{16} \) |
| 67 | \( 1 - 6 T + 166 T^{2} - 924 T^{3} + 11089 T^{4} - 78300 T^{5} + 988546 T^{6} - 7031886 T^{7} + 92397772 T^{8} - 7031886 p T^{9} + 988546 p^{2} T^{10} - 78300 p^{3} T^{11} + 11089 p^{4} T^{12} - 924 p^{5} T^{13} + 166 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 71 | \( 1 + 66 T^{2} - 5411 T^{4} - 1080 T^{5} - 5238 T^{6} + 2109672 T^{7} + 49128204 T^{8} + 2109672 p T^{9} - 5238 p^{2} T^{10} - 1080 p^{3} T^{11} - 5411 p^{4} T^{12} + 66 p^{6} T^{14} + p^{8} T^{16} \) |
| 73 | \( 1 - 352 T^{2} + 64540 T^{4} - 7783072 T^{6} + 668463622 T^{8} - 7783072 p^{2} T^{10} + 64540 p^{4} T^{12} - 352 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( ( 1 + 8 T + 184 T^{2} + 1256 T^{3} + 21022 T^{4} + 1256 p T^{5} + 184 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 - 472 T^{2} + 104380 T^{4} - 14464552 T^{6} + 1407855142 T^{8} - 14464552 p^{2} T^{10} + 104380 p^{4} T^{12} - 472 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( 1 - 24 T + 338 T^{2} - 3504 T^{3} + 22477 T^{4} - 143640 T^{5} + 783482 T^{6} - 3203040 T^{7} + 47062828 T^{8} - 3203040 p T^{9} + 783482 p^{2} T^{10} - 143640 p^{3} T^{11} + 22477 p^{4} T^{12} - 3504 p^{5} T^{13} + 338 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \) |
| 97 | \( 1 + 30 T + 746 T^{2} + 13380 T^{3} + 217333 T^{4} + 2990232 T^{5} + 37765718 T^{6} + 423036774 T^{7} + 4390187620 T^{8} + 423036774 p T^{9} + 37765718 p^{2} T^{10} + 2990232 p^{3} T^{11} + 217333 p^{4} T^{12} + 13380 p^{5} T^{13} + 746 p^{6} T^{14} + 30 p^{7} T^{15} + p^{8} T^{16} \) |
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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
−7.39259802992141814342508989946, −7.35917011631998250470400808053, −6.71788102904751183275109623967, −6.70790455546909363989073532389, −6.43985310246550613953770658146, −6.21463565686403357216804174598, −6.08625732330117526726608202729, −6.03340230897163970228686839793, −5.74789291303059430250809435651, −5.73154363345460667563288823068, −5.23599059623554073228555263819, −4.99175200570362950331213995770, −4.76562561577916403878006924316, −4.56804986037642860765581972549, −4.55811518716557162412556913308, −4.18542212044533859684091170510, −4.03098402667033651751282679777, −3.51165273711499983201429723159, −3.44960781111466495319585531492, −3.38957898435949272899290149241, −3.15024995636374636322778007186, −2.99578752465532891473641690173, −2.21990749613258928673505807673, −2.03773300000244614306454951812, −1.47694879281105926563825759706,
1.47694879281105926563825759706, 2.03773300000244614306454951812, 2.21990749613258928673505807673, 2.99578752465532891473641690173, 3.15024995636374636322778007186, 3.38957898435949272899290149241, 3.44960781111466495319585531492, 3.51165273711499983201429723159, 4.03098402667033651751282679777, 4.18542212044533859684091170510, 4.55811518716557162412556913308, 4.56804986037642860765581972549, 4.76562561577916403878006924316, 4.99175200570362950331213995770, 5.23599059623554073228555263819, 5.73154363345460667563288823068, 5.74789291303059430250809435651, 6.03340230897163970228686839793, 6.08625732330117526726608202729, 6.21463565686403357216804174598, 6.43985310246550613953770658146, 6.70790455546909363989073532389, 6.71788102904751183275109623967, 7.35917011631998250470400808053, 7.39259802992141814342508989946
Plot not available for L-functions of degree greater than 10.
