| L(s) = 1 | + 4·2-s + 4·3-s + 6·4-s − 2·5-s + 16·6-s + 6·9-s − 8·10-s − 6·11-s + 24·12-s − 4·13-s − 8·15-s − 15·16-s + 2·17-s + 24·18-s + 6·19-s − 12·20-s − 24·22-s − 4·23-s + 9·25-s − 16·26-s − 20·29-s − 32·30-s + 14·31-s − 24·32-s − 24·33-s + 8·34-s + 36·36-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 2.30·3-s + 3·4-s − 0.894·5-s + 6.53·6-s + 2·9-s − 2.52·10-s − 1.80·11-s + 6.92·12-s − 1.10·13-s − 2.06·15-s − 3.75·16-s + 0.485·17-s + 5.65·18-s + 1.37·19-s − 2.68·20-s − 5.11·22-s − 0.834·23-s + 9/5·25-s − 3.13·26-s − 3.71·29-s − 5.84·30-s + 2.51·31-s − 4.24·32-s − 4.17·33-s + 1.37·34-s + 6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(14.62024523\) |
| \(L(\frac12)\) |
\(\approx\) |
\(14.62024523\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T + T^{2} )^{4} \) |
| 3 | \( ( 1 - T + T^{2} )^{4} \) |
| 7 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | \( ( 1 + T + T^{2} )^{4} \) |
| good | 5 | \( 1 + 2 T - p T^{2} - 18 T^{3} - 3 T^{4} + 12 p T^{5} + 162 T^{6} - 56 T^{7} - 966 T^{8} - 56 p T^{9} + 162 p^{2} T^{10} + 12 p^{4} T^{11} - 3 p^{4} T^{12} - 18 p^{5} T^{13} - p^{7} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 11 | \( 1 + 6 T + 14 T^{2} + 76 T^{3} + 273 T^{4} + 408 T^{5} + 3314 T^{6} + 11610 T^{7} + 15852 T^{8} + 11610 p T^{9} + 3314 p^{2} T^{10} + 408 p^{3} T^{11} + 273 p^{4} T^{12} + 76 p^{5} T^{13} + 14 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 13 | \( ( 1 + 2 T + 18 T^{2} - 34 T^{3} + 74 T^{4} - 34 p T^{5} + 18 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 17 | \( 1 - 2 T - 39 T^{2} - 46 T^{3} + 987 T^{4} + 2088 T^{5} - 10772 T^{6} - 21880 T^{7} + 90306 T^{8} - 21880 p T^{9} - 10772 p^{2} T^{10} + 2088 p^{3} T^{11} + 987 p^{4} T^{12} - 46 p^{5} T^{13} - 39 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 19 | \( 1 - 6 T - 6 T^{2} + 272 T^{3} - 978 T^{4} - 1942 T^{5} + 19960 T^{6} - 20514 T^{7} - 208613 T^{8} - 20514 p T^{9} + 19960 p^{2} T^{10} - 1942 p^{3} T^{11} - 978 p^{4} T^{12} + 272 p^{5} T^{13} - 6 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 29 | \( ( 1 + 10 T + 106 T^{2} + 744 T^{3} + 4655 T^{4} + 744 p T^{5} + 106 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( 1 - 14 T + 67 T^{2} - 42 T^{3} - 37 p T^{4} + 11200 T^{5} - 58686 T^{6} + 103516 T^{7} + 240506 T^{8} + 103516 p T^{9} - 58686 p^{2} T^{10} + 11200 p^{3} T^{11} - 37 p^{5} T^{12} - 42 p^{5} T^{13} + 67 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \) |
| 37 | \( 1 + 6 T - 90 T^{2} - 544 T^{3} + 5358 T^{4} + 26534 T^{5} - 210680 T^{6} - 420018 T^{7} + 8204035 T^{8} - 420018 p T^{9} - 210680 p^{2} T^{10} + 26534 p^{3} T^{11} + 5358 p^{4} T^{12} - 544 p^{5} T^{13} - 90 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 41 | \( ( 1 - 10 T + 154 T^{2} - 1006 T^{3} + 9210 T^{4} - 1006 p T^{5} + 154 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 2 T + p T^{2} )^{8} \) |
| 47 | \( 1 - 10 T - 9 T^{2} - 374 T^{3} + 5061 T^{4} + 8952 T^{5} + 88306 T^{6} - 937892 T^{7} - 5886318 T^{8} - 937892 p T^{9} + 88306 p^{2} T^{10} + 8952 p^{3} T^{11} + 5061 p^{4} T^{12} - 374 p^{5} T^{13} - 9 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) |
| 53 | \( 1 + 6 T - 177 T^{2} - 582 T^{3} + 22841 T^{4} + 44004 T^{5} - 1844130 T^{6} - 752736 T^{7} + 116782146 T^{8} - 752736 p T^{9} - 1844130 p^{2} T^{10} + 44004 p^{3} T^{11} + 22841 p^{4} T^{12} - 582 p^{5} T^{13} - 177 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 59 | \( 1 + 24 T + 243 T^{2} + 1800 T^{3} + 11357 T^{4} + 2688 T^{5} - 881298 T^{6} - 10799088 T^{7} - 90601122 T^{8} - 10799088 p T^{9} - 881298 p^{2} T^{10} + 2688 p^{3} T^{11} + 11357 p^{4} T^{12} + 1800 p^{5} T^{13} + 243 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \) |
| 61 | \( 1 - 28 T + 296 T^{2} - 2072 T^{3} + 23634 T^{4} - 257572 T^{5} + 1841408 T^{6} - 14807268 T^{7} + 133853251 T^{8} - 14807268 p T^{9} + 1841408 p^{2} T^{10} - 257572 p^{3} T^{11} + 23634 p^{4} T^{12} - 2072 p^{5} T^{13} + 296 p^{6} T^{14} - 28 p^{7} T^{15} + p^{8} T^{16} \) |
| 67 | \( 1 + 28 T + 364 T^{2} + 2968 T^{3} + 15018 T^{4} - 3612 T^{5} - 1234800 T^{6} - 17854452 T^{7} - 168746573 T^{8} - 17854452 p T^{9} - 1234800 p^{2} T^{10} - 3612 p^{3} T^{11} + 15018 p^{4} T^{12} + 2968 p^{5} T^{13} + 364 p^{6} T^{14} + 28 p^{7} T^{15} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 16 T + 351 T^{2} - 3474 T^{3} + 39778 T^{4} - 3474 p T^{5} + 351 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( 1 + 6 T - 153 T^{2} + 242 T^{3} + 17733 T^{4} - 66892 T^{5} - 876038 T^{6} + 2899416 T^{7} + 29190250 T^{8} + 2899416 p T^{9} - 876038 p^{2} T^{10} - 66892 p^{3} T^{11} + 17733 p^{4} T^{12} + 242 p^{5} T^{13} - 153 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 79 | \( 1 - 4 T - 74 T^{2} - 952 T^{3} + 6333 T^{4} + 73632 T^{5} + 1038942 T^{6} - 7554540 T^{7} - 71938196 T^{8} - 7554540 p T^{9} + 1038942 p^{2} T^{10} + 73632 p^{3} T^{11} + 6333 p^{4} T^{12} - 952 p^{5} T^{13} - 74 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 83 | \( ( 1 - 14 T + 289 T^{2} - 2310 T^{3} + 30488 T^{4} - 2310 p T^{5} + 289 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 89 | \( 1 - 14 T - 74 T^{2} + 1008 T^{3} + 10238 T^{4} - 43358 T^{5} - 843768 T^{6} + 5469170 T^{7} - 24425981 T^{8} + 5469170 p T^{9} - 843768 p^{2} T^{10} - 43358 p^{3} T^{11} + 10238 p^{4} T^{12} + 1008 p^{5} T^{13} - 74 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \) |
| 97 | \( ( 1 + 69 T^{2} + 112 T^{3} + 18456 T^{4} + 112 p T^{5} + 69 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−4.27970103985676189746838587581, −4.15678081309367158709916973171, −3.95882541029332949109262696145, −3.82072166885479504917372188920, −3.80958252426117253459501724341, −3.71086393408800221329302336783, −3.40250737749182928143221509044, −3.37183496764510524353310868538, −3.29153890395277337995721667323, −3.25205322202905297379961195260, −3.13751036255344832119267435600, −2.75300824349481686765708202252, −2.59865715959141088111168215992, −2.59124759734991352928210211836, −2.57397531417956955520802278409, −2.45463746133463861279947110657, −2.29436673472892591795686354745, −2.14457874918099353107650352071, −1.70138446765096994848754670551, −1.70125508240432751643241372498, −1.52641132320620995268383451907, −0.883687605193698128324155961773, −0.873936767804901149667619774306, −0.60860618292638880827794247229, −0.22152349875347294722002976660,
0.22152349875347294722002976660, 0.60860618292638880827794247229, 0.873936767804901149667619774306, 0.883687605193698128324155961773, 1.52641132320620995268383451907, 1.70125508240432751643241372498, 1.70138446765096994848754670551, 2.14457874918099353107650352071, 2.29436673472892591795686354745, 2.45463746133463861279947110657, 2.57397531417956955520802278409, 2.59124759734991352928210211836, 2.59865715959141088111168215992, 2.75300824349481686765708202252, 3.13751036255344832119267435600, 3.25205322202905297379961195260, 3.29153890395277337995721667323, 3.37183496764510524353310868538, 3.40250737749182928143221509044, 3.71086393408800221329302336783, 3.80958252426117253459501724341, 3.82072166885479504917372188920, 3.95882541029332949109262696145, 4.15678081309367158709916973171, 4.27970103985676189746838587581
Plot not available for L-functions of degree greater than 10.
