| L(s) = 1 | + 2·4-s + 2·9-s + 16-s − 6·19-s + 18·29-s − 4·31-s + 4·36-s − 16·41-s − 15·49-s + 34·59-s − 24·61-s − 2·64-s + 36·71-s − 12·76-s − 76·79-s + 81-s − 34·89-s + 52·101-s + 96·109-s + 36·116-s + 10·121-s − 8·124-s + 127-s + 131-s + 137-s + 139-s + 2·144-s + ⋯ |
| L(s) = 1 | + 4-s + 2/3·9-s + 1/4·16-s − 1.37·19-s + 3.34·29-s − 0.718·31-s + 2/3·36-s − 2.49·41-s − 2.14·49-s + 4.42·59-s − 3.07·61-s − 1/4·64-s + 4.27·71-s − 1.37·76-s − 8.55·79-s + 1/9·81-s − 3.60·89-s + 5.17·101-s + 9.19·109-s + 3.34·116-s + 0.909·121-s − 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1/6·144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.73982366\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.73982366\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 3 | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 5 | \( 1 \) |
| 13 | \( 1 + 25 T^{2} + 456 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \) |
| good | 7 | \( 1 + 15 T^{2} + 109 T^{4} + 270 T^{6} + 30 T^{8} + 270 p^{2} T^{10} + 109 p^{4} T^{12} + 15 p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( ( 1 - 9 T + 38 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2}( 1 + 9 T + 38 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | \( 1 + 32 T^{2} + 258 T^{4} + 6016 T^{6} + 207299 T^{8} + 6016 p^{2} T^{10} + 258 p^{4} T^{12} + 32 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( ( 1 + 3 T - 27 T^{2} - 6 T^{3} + 764 T^{4} - 6 p T^{5} - 27 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 23 | \( 1 + 11 T^{2} - 623 T^{4} - 3454 T^{6} + 209686 T^{8} - 3454 p^{2} T^{10} - 623 p^{4} T^{12} + 11 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( ( 1 - 9 T + 7 T^{2} - 144 T^{3} + 2286 T^{4} - 144 p T^{5} + 7 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + T + 24 T^{2} + p T^{3} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 + 57 T^{2} + 1880 T^{4} + 57 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 8 T + 34 T^{2} - 416 T^{3} - 3405 T^{4} - 416 p T^{5} + 34 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 + 151 T^{2} + 13509 T^{4} + 844694 T^{6} + 40763414 T^{8} + 844694 p^{2} T^{10} + 13509 p^{4} T^{12} + 151 p^{6} T^{14} + p^{8} T^{16} \) |
| 47 | \( ( 1 - 45 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 - 119 T^{2} + 8440 T^{4} - 119 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 17 T + 103 T^{2} - 1156 T^{3} + 14064 T^{4} - 1156 p T^{5} + 103 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( 1 + 60 T^{2} + 3514 T^{4} - 533520 T^{6} - 36304125 T^{8} - 533520 p^{2} T^{10} + 3514 p^{4} T^{12} + 60 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 18 T + 118 T^{2} - 1152 T^{3} + 14391 T^{4} - 1152 p T^{5} + 118 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 32 T^{2} + 5406 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 19 T + 244 T^{2} + 19 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 280 T^{2} + 32766 T^{4} - 280 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 17 T + 145 T^{2} - 578 T^{3} - 12906 T^{4} - 578 p T^{5} + 145 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 97 | \( 1 + 336 T^{2} + 66466 T^{4} + 9277632 T^{6} + 1004977155 T^{8} + 9277632 p^{2} T^{10} + 66466 p^{4} T^{12} + 336 p^{6} T^{14} + p^{8} T^{16} \) |
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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
−3.85511980865007324380485656128, −3.72622862256365874806523652855, −3.52711322970781169105374422348, −3.34632445042360923801495605191, −3.34142827044178608614368269369, −3.28412584279150506615181350224, −3.12854865285485686396537603424, −2.98872391811124551503370218122, −2.89030819897034164576110448280, −2.67084618996389664672651651141, −2.54089095482206124678955399492, −2.51184252108664876834001200849, −2.41755616200565892754369870735, −2.03668451704657700972991029318, −1.94957781136048794268062162836, −1.82172302761490790217750419846, −1.80165861307163511130017118014, −1.70044406178999949016673509209, −1.42994971897234167969885681697, −1.41732411732129025527618781081, −1.01106139903692672474078142961, −0.797729931412272096327385139835, −0.70322667241907149127332760433, −0.42088625231916027229762144643, −0.29714564399343292272137790308,
0.29714564399343292272137790308, 0.42088625231916027229762144643, 0.70322667241907149127332760433, 0.797729931412272096327385139835, 1.01106139903692672474078142961, 1.41732411732129025527618781081, 1.42994971897234167969885681697, 1.70044406178999949016673509209, 1.80165861307163511130017118014, 1.82172302761490790217750419846, 1.94957781136048794268062162836, 2.03668451704657700972991029318, 2.41755616200565892754369870735, 2.51184252108664876834001200849, 2.54089095482206124678955399492, 2.67084618996389664672651651141, 2.89030819897034164576110448280, 2.98872391811124551503370218122, 3.12854865285485686396537603424, 3.28412584279150506615181350224, 3.34142827044178608614368269369, 3.34632445042360923801495605191, 3.52711322970781169105374422348, 3.72622862256365874806523652855, 3.85511980865007324380485656128
Plot not available for L-functions of degree greater than 10.
