| L(s) = 1 | − 3·9-s + 12·13-s + 24·29-s + 22·37-s + 18·41-s − 15·49-s + 32·53-s + 22·61-s − 50·73-s + 6·81-s + 10·89-s − 46·97-s + 44·101-s + 22·109-s − 4·113-s − 36·117-s − 39·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 38·169-s + ⋯ |
| L(s) = 1 | − 9-s + 3.32·13-s + 4.45·29-s + 3.61·37-s + 2.81·41-s − 2.14·49-s + 4.39·53-s + 2.81·61-s − 5.85·73-s + 2/3·81-s + 1.05·89-s − 4.67·97-s + 4.37·101-s + 2.10·109-s − 0.376·113-s − 3.32·117-s − 3.54·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.92·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(18.39621204\) |
| \(L(\frac12)\) |
\(\approx\) |
\(18.39621204\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} + p T^{4} + 5 T^{6} + 56 T^{8} + 5 p^{2} T^{10} + p^{5} T^{12} + p^{7} T^{14} + p^{8} T^{16} \) |
| 7 | \( 1 + 15 T^{2} + 174 T^{4} + 1140 T^{6} + 8601 T^{8} + 1140 p^{2} T^{10} + 174 p^{4} T^{12} + 15 p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( 1 + 39 T^{2} + 838 T^{4} + 13252 T^{6} + 163289 T^{8} + 13252 p^{2} T^{10} + 838 p^{4} T^{12} + 39 p^{6} T^{14} + p^{8} T^{16} \) |
| 13 | \( ( 1 - 6 T + 35 T^{2} - 12 p T^{3} + 721 T^{4} - 12 p^{2} T^{5} + 35 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 35 T^{2} + 20 T^{3} + 788 T^{4} + 20 p T^{5} + 35 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( 1 + 72 T^{2} + 2633 T^{4} + 70424 T^{6} + 1505265 T^{8} + 70424 p^{2} T^{10} + 2633 p^{4} T^{12} + 72 p^{6} T^{14} + p^{8} T^{16} \) |
| 23 | \( 1 + 93 T^{2} + 4583 T^{4} + 157855 T^{6} + 4136376 T^{8} + 157855 p^{2} T^{10} + 4583 p^{4} T^{12} + 93 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( ( 1 - 12 T + 137 T^{2} - 974 T^{3} + 6284 T^{4} - 974 p T^{5} + 137 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( 1 + 219 T^{2} + 21723 T^{4} + 1275717 T^{6} + 48544344 T^{8} + 1275717 p^{2} T^{10} + 21723 p^{4} T^{12} + 219 p^{6} T^{14} + p^{8} T^{16} \) |
| 37 | \( ( 1 - 11 T + 137 T^{2} - 941 T^{3} + 7080 T^{4} - 941 p T^{5} + 137 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 9 T + 158 T^{2} - 1012 T^{3} + 9689 T^{4} - 1012 p T^{5} + 158 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 + 110 T^{2} + 8509 T^{4} + 431950 T^{6} + 20207356 T^{8} + 431950 p^{2} T^{10} + 8509 p^{4} T^{12} + 110 p^{6} T^{14} + p^{8} T^{16} \) |
| 47 | \( 1 + 200 T^{2} + 20289 T^{4} + 1447840 T^{6} + 78247401 T^{8} + 1447840 p^{2} T^{10} + 20289 p^{4} T^{12} + 200 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( ( 1 - 16 T + 153 T^{2} - 1460 T^{3} + 13101 T^{4} - 1460 p T^{5} + 153 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 59 | \( 1 + 276 T^{2} + 39893 T^{4} + 3851248 T^{6} + 265932489 T^{8} + 3851248 p^{2} T^{10} + 39893 p^{4} T^{12} + 276 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( ( 1 - 11 T + 3 p T^{2} - 1033 T^{3} + 12264 T^{4} - 1033 p T^{5} + 3 p^{3} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 + 455 T^{2} + 93499 T^{4} + 11487105 T^{6} + 933230216 T^{8} + 11487105 p^{2} T^{10} + 93499 p^{4} T^{12} + 455 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( 1 - 126 T^{2} + 24173 T^{4} - 1916498 T^{6} + 192814644 T^{8} - 1916498 p^{2} T^{10} + 24173 p^{4} T^{12} - 126 p^{6} T^{14} + p^{8} T^{16} \) |
| 73 | \( ( 1 + 25 T + 415 T^{2} + 4805 T^{3} + 45808 T^{4} + 4805 p T^{5} + 415 p^{2} T^{6} + 25 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( 1 + 86 T^{2} - 5987 T^{4} + 11758 T^{6} + 86984524 T^{8} + 11758 p^{2} T^{10} - 5987 p^{4} T^{12} + 86 p^{6} T^{14} + p^{8} T^{16} \) |
| 83 | \( 1 + 88 T^{2} - 3332 T^{4} - 280920 T^{6} + 13185446 T^{8} - 280920 p^{2} T^{10} - 3332 p^{4} T^{12} + 88 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 5 T + 111 T^{2} + 565 T^{3} + 1916 T^{4} + 565 p T^{5} + 111 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 23 T + 407 T^{2} + 5275 T^{3} + 59896 T^{4} + 5275 p T^{5} + 407 p^{2} T^{6} + 23 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−3.45803555726512465564592587075, −3.24314311514695311367594896349, −3.15243835068911931001873156799, −3.04051595403143129992151886704, −2.93929047057528468361837332670, −2.88797980316666155880790882959, −2.84779251651863862793170778559, −2.84496666610447363824567513150, −2.65852113603607805088322350155, −2.32603467491056108864746141048, −2.26414395251734939994870053320, −2.22287009386814475147772705796, −2.18731677050440668034513482693, −2.01316778961781827569580989235, −1.83675006813623762523617089562, −1.54592617374951891254445053190, −1.29724632815750646841635543399, −1.24075030938407605417984773499, −1.13194789139611028855845557062, −1.03620450248980508493560701810, −0.944781614976522286750052035697, −0.884714000483124070449202194840, −0.71025139065869643104829654737, −0.33923374556720775410280574529, −0.26263975685922825557014063265,
0.26263975685922825557014063265, 0.33923374556720775410280574529, 0.71025139065869643104829654737, 0.884714000483124070449202194840, 0.944781614976522286750052035697, 1.03620450248980508493560701810, 1.13194789139611028855845557062, 1.24075030938407605417984773499, 1.29724632815750646841635543399, 1.54592617374951891254445053190, 1.83675006813623762523617089562, 2.01316778961781827569580989235, 2.18731677050440668034513482693, 2.22287009386814475147772705796, 2.26414395251734939994870053320, 2.32603467491056108864746141048, 2.65852113603607805088322350155, 2.84496666610447363824567513150, 2.84779251651863862793170778559, 2.88797980316666155880790882959, 2.93929047057528468361837332670, 3.04051595403143129992151886704, 3.15243835068911931001873156799, 3.24314311514695311367594896349, 3.45803555726512465564592587075
Plot not available for L-functions of degree greater than 10.
