| L(s) = 1 | − 4·11-s + 20·19-s − 44·59-s − 12·61-s − 64·71-s − 12·79-s + 81-s + 36·89-s + 12·101-s + 48·109-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 1.20·11-s + 4.58·19-s − 5.72·59-s − 1.53·61-s − 7.59·71-s − 1.35·79-s + 1/9·81-s + 3.81·89-s + 1.19·101-s + 4.59·109-s + 4·121-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 + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.038411086\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.038411086\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T^{4} + T^{8} \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 23 T^{4} + p^{4} T^{8} \) |
| good | 11 | \( ( 1 + 2 T - 16 T^{2} - 4 T^{3} + 235 T^{4} - 4 p T^{5} - 16 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 + 287 T^{4} + p^{4} T^{8} )^{2} \) |
| 17 | \( 1 - 60 T^{2} + 2042 T^{4} - 50520 T^{6} + 972243 T^{8} - 50520 p^{2} T^{10} + 2042 p^{4} T^{12} - 60 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( ( 1 - 10 T + 49 T^{2} - 130 T^{3} + 340 T^{4} - 130 p T^{5} + 49 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 23 | \( 1 + 96 T^{2} + 4574 T^{4} + 144192 T^{6} + 3601251 T^{8} + 144192 p^{2} T^{10} + 4574 p^{4} T^{12} + 96 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 + 58 T^{2} + 2403 T^{4} + 58 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( 1 - 216 T^{2} + 22057 T^{4} - 1405080 T^{6} + 61731552 T^{8} - 1405080 p^{2} T^{10} + 22057 p^{4} T^{12} - 216 p^{6} T^{14} + p^{8} T^{16} \) |
| 41 | \( ( 1 - 108 T^{2} + 6170 T^{4} - 108 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 + 868 T^{4} - 3308250 T^{8} + 868 p^{4} T^{12} + p^{8} T^{16} \) |
| 47 | \( 1 + 180 T^{2} + 16154 T^{4} + 963720 T^{6} + 47642835 T^{8} + 963720 p^{2} T^{10} + 16154 p^{4} T^{12} + 180 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( 1 - 84 T^{2} + 5642 T^{4} - 276360 T^{6} + 9540387 T^{8} - 276360 p^{2} T^{10} + 5642 p^{4} T^{12} - 84 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( ( 1 + 22 T + 248 T^{2} + 44 p T^{3} + 401 p T^{4} + 44 p^{2} T^{5} + 248 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 6 T + 73 T^{2} + 6 p T^{3} + 12 p T^{4} + 6 p^{2} T^{5} + 73 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 + 1753 T^{4} - 17078112 T^{8} + 1753 p^{4} T^{12} + p^{8} T^{16} \) |
| 71 | \( ( 1 + 8 T + p T^{2} )^{8} \) |
| 73 | \( 1 + 360 T^{2} + 64489 T^{4} + 7664040 T^{6} + 655036080 T^{8} + 7664040 p^{2} T^{10} + 64489 p^{4} T^{12} + 360 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( ( 1 + 6 T + 157 T^{2} + 870 T^{3} + 15732 T^{4} + 870 p T^{5} + 157 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 + 20092 T^{4} + 185593446 T^{8} + 20092 p^{4} T^{12} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 18 T + 68 T^{2} - 1404 T^{3} + 28779 T^{4} - 1404 p T^{5} + 68 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 97 | \( 1 + 2446 T^{4} + 12136179 T^{8} + 2446 p^{4} T^{12} + p^{8} T^{16} \) |
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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
−3.60864044835479833014016074296, −3.49671590572088552863426183028, −3.47799716979080653263062776663, −3.38325956234557770621002199820, −3.31602861385427814564882377042, −3.16823334207641111538028031002, −3.16625780137755999327350384651, −2.91159804583901531889768829041, −2.88119116509988824600101034569, −2.88056829542300328972157043928, −2.74816985693648907780965411537, −2.35713416241932351311526819264, −2.18845709782407283692175192225, −2.18711791735855127077315016351, −2.09253516839301466508510639566, −1.78189942044811172211619857868, −1.61860235675289328476401485984, −1.53518587654110092093451179492, −1.28530891593949684908127401729, −1.24383611088961721814506272273, −1.15442134288543521610766636677, −0.962984353230938719272106148768, −0.48119969604293561772058938689, −0.43661321582566650251064518383, −0.18200231802092872000545437765,
0.18200231802092872000545437765, 0.43661321582566650251064518383, 0.48119969604293561772058938689, 0.962984353230938719272106148768, 1.15442134288543521610766636677, 1.24383611088961721814506272273, 1.28530891593949684908127401729, 1.53518587654110092093451179492, 1.61860235675289328476401485984, 1.78189942044811172211619857868, 2.09253516839301466508510639566, 2.18711791735855127077315016351, 2.18845709782407283692175192225, 2.35713416241932351311526819264, 2.74816985693648907780965411537, 2.88056829542300328972157043928, 2.88119116509988824600101034569, 2.91159804583901531889768829041, 3.16625780137755999327350384651, 3.16823334207641111538028031002, 3.31602861385427814564882377042, 3.38325956234557770621002199820, 3.47799716979080653263062776663, 3.49671590572088552863426183028, 3.60864044835479833014016074296
Plot not available for L-functions of degree greater than 10.
