| L(s) = 1 | + 8·2-s + 36·4-s + 10·7-s + 120·8-s − 4·9-s + 80·14-s + 330·16-s − 32·18-s + 360·28-s − 6·29-s + 792·32-s − 144·36-s + 40·37-s − 6·47-s + 37·49-s + 1.20e3·56-s − 48·58-s + 10·61-s − 40·63-s + 1.71e3·64-s + 32·67-s − 480·72-s + 26·73-s + 320·74-s − 4·79-s − 8·81-s + 48·83-s + ⋯ |
| L(s) = 1 | + 5.65·2-s + 18·4-s + 3.77·7-s + 42.4·8-s − 4/3·9-s + 21.3·14-s + 82.5·16-s − 7.54·18-s + 68.0·28-s − 1.11·29-s + 140.·32-s − 24·36-s + 6.57·37-s − 0.875·47-s + 37/7·49-s + 160.·56-s − 6.30·58-s + 1.28·61-s − 5.03·63-s + 214.5·64-s + 3.90·67-s − 56.5·72-s + 3.04·73-s + 37.1·74-s − 0.450·79-s − 8/9·81-s + 5.26·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5955.294557\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5955.294557\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T )^{8} \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 4 T^{2} + 8 p T^{4} + 64 T^{6} + 262 T^{8} + 64 p^{2} T^{10} + 8 p^{5} T^{12} + 4 p^{6} T^{14} + p^{8} T^{16} \) |
| 7 | \( ( 1 - 5 T + 19 T^{2} - 68 T^{3} + 220 T^{4} - 68 p T^{5} + 19 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 11 | \( 1 + 47 T^{2} + 1081 T^{4} + 16586 T^{6} + 199906 T^{8} + 16586 p^{2} T^{10} + 1081 p^{4} T^{12} + 47 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( 1 + 77 T^{2} + 2554 T^{4} + 51383 T^{6} + 860374 T^{8} + 51383 p^{2} T^{10} + 2554 p^{4} T^{12} + 77 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( 1 + 107 T^{2} + 5677 T^{4} + 189206 T^{6} + 4307434 T^{8} + 189206 p^{2} T^{10} + 5677 p^{4} T^{12} + 107 p^{6} T^{14} + p^{8} T^{16} \) |
| 23 | \( 1 + 140 T^{2} + 9076 T^{4} + 362564 T^{6} + 9922870 T^{8} + 362564 p^{2} T^{10} + 9076 p^{4} T^{12} + 140 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( ( 1 + 3 T + 98 T^{2} + 201 T^{3} + 3978 T^{4} + 201 p T^{5} + 98 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( 1 + 188 T^{2} + 16924 T^{4} + 941624 T^{6} + 35207710 T^{8} + 941624 p^{2} T^{10} + 16924 p^{4} T^{12} + 188 p^{6} T^{14} + p^{8} T^{16} \) |
| 37 | \( ( 1 - 20 T + 268 T^{2} - 2438 T^{3} + 17203 T^{4} - 2438 p T^{5} + 268 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( 1 + 125 T^{2} + 8578 T^{4} + 400067 T^{6} + 16651690 T^{8} + 400067 p^{2} T^{10} + 8578 p^{4} T^{12} + 125 p^{6} T^{14} + p^{8} T^{16} \) |
| 43 | \( 1 + 164 T^{2} + 16516 T^{4} + 1105532 T^{6} + 55599574 T^{8} + 1105532 p^{2} T^{10} + 16516 p^{4} T^{12} + 164 p^{6} T^{14} + p^{8} T^{16} \) |
| 47 | \( ( 1 + 3 T + 71 T^{2} + 156 T^{3} + 2160 T^{4} + 156 p T^{5} + 71 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 53 | \( 1 + 68 T^{2} + 7318 T^{4} + 456764 T^{6} + 29553859 T^{8} + 456764 p^{2} T^{10} + 7318 p^{4} T^{12} + 68 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( 1 + 260 T^{2} + 32260 T^{4} + 2631068 T^{6} + 168803926 T^{8} + 2631068 p^{2} T^{10} + 32260 p^{4} T^{12} + 260 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( ( 1 - 5 T - 2 T^{2} + 109 T^{3} - 2324 T^{4} + 109 p T^{5} - 2 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 4 T + p T^{2} )^{8} \) |
| 71 | \( 1 + 248 T^{2} + 38620 T^{4} + 4083272 T^{6} + 333935494 T^{8} + 4083272 p^{2} T^{10} + 38620 p^{4} T^{12} + 248 p^{6} T^{14} + p^{8} T^{16} \) |
| 73 | \( ( 1 - 13 T + 178 T^{2} - 1549 T^{3} + 13924 T^{4} - 1549 p T^{5} + 178 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 2 T + 100 T^{2} - 28 T^{3} + 4702 T^{4} - 28 p T^{5} + 100 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 - 24 T + 410 T^{2} - 4374 T^{3} + 44538 T^{4} - 4374 p T^{5} + 410 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 89 | \( 1 + 287 T^{2} + 52153 T^{4} + 6263882 T^{6} + 627405838 T^{8} + 6263882 p^{2} T^{10} + 52153 p^{4} T^{12} + 287 p^{6} T^{14} + p^{8} T^{16} \) |
| 97 | \( ( 1 - 4 T + 328 T^{2} - 778 T^{3} + 44242 T^{4} - 778 p T^{5} + 328 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−3.40251885251902750395397785591, −3.01209461257861565448115468642, −2.85527657220175220056295335903, −2.82505257962639301020585379677, −2.73372507792889892575465246662, −2.68295276967053948869209719306, −2.59964969566727510870064507713, −2.53251129064090698084188279370, −2.50763766101011384785971447932, −2.19245748169349060747345960695, −2.05890509569967555760889545293, −1.98940335409441866662335386445, −1.96737691703227052298493723139, −1.94280067184997590482764577598, −1.87990664474293306194574173949, −1.77525471794139325960679301072, −1.49374851431072334689694113732, −1.26780077634953221649012663837, −1.14840538172659273561792743760, −1.13286620092807746465145575181, −0.860119319266584034013167392171, −0.848821126778620410789214071315, −0.809053596017839560377571985994, −0.46532637909065311232229640695, −0.32353281244417978609120065173,
0.32353281244417978609120065173, 0.46532637909065311232229640695, 0.809053596017839560377571985994, 0.848821126778620410789214071315, 0.860119319266584034013167392171, 1.13286620092807746465145575181, 1.14840538172659273561792743760, 1.26780077634953221649012663837, 1.49374851431072334689694113732, 1.77525471794139325960679301072, 1.87990664474293306194574173949, 1.94280067184997590482764577598, 1.96737691703227052298493723139, 1.98940335409441866662335386445, 2.05890509569967555760889545293, 2.19245748169349060747345960695, 2.50763766101011384785971447932, 2.53251129064090698084188279370, 2.59964969566727510870064507713, 2.68295276967053948869209719306, 2.73372507792889892575465246662, 2.82505257962639301020585379677, 2.85527657220175220056295335903, 3.01209461257861565448115468642, 3.40251885251902750395397785591
Plot not available for L-functions of degree greater than 10.
