| L(s) = 1 | − 4·2-s − 6·3-s + 4·4-s + 2·5-s + 24·6-s + 8·8-s + 18·9-s − 8·10-s + 6·11-s − 24·12-s − 2·13-s − 12·15-s − 30·16-s − 16·17-s − 72·18-s + 14·19-s + 8·20-s − 24·22-s + 14·23-s − 48·24-s + 8·25-s + 8·26-s − 38·27-s + 48·30-s + 2·31-s + 36·32-s − 36·33-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 3.46·3-s + 2·4-s + 0.894·5-s + 9.79·6-s + 2.82·8-s + 6·9-s − 2.52·10-s + 1.80·11-s − 6.92·12-s − 0.554·13-s − 3.09·15-s − 7.5·16-s − 3.88·17-s − 16.9·18-s + 3.21·19-s + 1.78·20-s − 5.11·22-s + 2.91·23-s − 9.79·24-s + 8/5·25-s + 1.56·26-s − 7.31·27-s + 8.76·30-s + 0.359·31-s + 6.36·32-s − 6.26·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.02616861464\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02616861464\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 2 T - 4 T^{2} - 6 T^{3} + 62 T^{4} - 6 p T^{5} - 4 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | \( 1 + 2 T - 2 p T^{3} - 330 T^{4} - 2 p^{2} T^{5} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| good | 2 | \( ( 1 + p T + p^{2} T^{2} + p^{2} T^{3} + 7 T^{4} + p^{3} T^{5} + p^{4} T^{6} + p^{4} T^{7} + p^{4} T^{8} )^{2} \) |
| 3 | \( 1 + 2 p T + 2 p^{2} T^{2} + 38 T^{3} + 68 T^{4} + 122 T^{5} + 230 T^{6} + 418 T^{7} + 730 T^{8} + 418 p T^{9} + 230 p^{2} T^{10} + 122 p^{3} T^{11} + 68 p^{4} T^{12} + 38 p^{5} T^{13} + 2 p^{8} T^{14} + 2 p^{8} T^{15} + p^{8} T^{16} \) |
| 7 | \( 1 - 16 T^{2} + 4 p^{2} T^{4} - 1824 T^{6} + 15206 T^{8} - 1824 p^{2} T^{10} + 4 p^{6} T^{12} - 16 p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( 1 - 6 T + 18 T^{2} - 74 T^{3} + 480 T^{4} - 1902 T^{5} + 5510 T^{6} - 22402 T^{7} + 90754 T^{8} - 22402 p T^{9} + 5510 p^{2} T^{10} - 1902 p^{3} T^{11} + 480 p^{4} T^{12} - 74 p^{5} T^{13} + 18 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 17 | \( 1 + 16 T + 128 T^{2} + 808 T^{3} + 4588 T^{4} + 1320 p T^{5} + 98208 T^{6} + 412960 T^{7} + 1709478 T^{8} + 412960 p T^{9} + 98208 p^{2} T^{10} + 1320 p^{4} T^{11} + 4588 p^{4} T^{12} + 808 p^{5} T^{13} + 128 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \) |
| 19 | \( 1 - 14 T + 98 T^{2} - 662 T^{3} + 4880 T^{4} - 28162 T^{5} + 135150 T^{6} - 689194 T^{7} + 3344898 T^{8} - 689194 p T^{9} + 135150 p^{2} T^{10} - 28162 p^{3} T^{11} + 4880 p^{4} T^{12} - 662 p^{5} T^{13} + 98 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \) |
| 23 | \( 1 - 14 T + 98 T^{2} - 610 T^{3} + 3540 T^{4} - 16574 T^{5} + 71166 T^{6} - 303274 T^{7} + 1330874 T^{8} - 303274 p T^{9} + 71166 p^{2} T^{10} - 16574 p^{3} T^{11} + 3540 p^{4} T^{12} - 610 p^{5} T^{13} + 98 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \) |
| 29 | \( 1 - 188 T^{2} + 16572 T^{4} - 885220 T^{6} + 31234358 T^{8} - 885220 p^{2} T^{10} + 16572 p^{4} T^{12} - 188 p^{6} T^{14} + p^{8} T^{16} \) |
| 31 | \( 1 - 2 T + 2 T^{2} - 30 T^{3} + 1184 T^{4} - 94 p T^{5} + 3910 T^{6} - 89742 T^{7} + 2057506 T^{8} - 89742 p T^{9} + 3910 p^{2} T^{10} - 94 p^{4} T^{11} + 1184 p^{4} T^{12} - 30 p^{5} T^{13} + 2 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 37 | \( 1 - 156 T^{2} + 13708 T^{4} - 811972 T^{6} + 34945238 T^{8} - 811972 p^{2} T^{10} + 13708 p^{4} T^{12} - 156 p^{6} T^{14} + p^{8} T^{16} \) |
| 41 | \( 1 - 16 T + 128 T^{2} - 744 T^{3} + 5852 T^{4} - 58120 T^{5} + 457632 T^{6} - 2817600 T^{7} + 17080198 T^{8} - 2817600 p T^{9} + 457632 p^{2} T^{10} - 58120 p^{3} T^{11} + 5852 p^{4} T^{12} - 744 p^{5} T^{13} + 128 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) |
| 43 | \( 1 - 6 T + 18 T^{2} + 2 p T^{3} + 596 T^{4} - 17470 T^{5} + 97790 T^{6} - 211786 T^{7} - 2781606 T^{8} - 211786 p T^{9} + 97790 p^{2} T^{10} - 17470 p^{3} T^{11} + 596 p^{4} T^{12} + 2 p^{6} T^{13} + 18 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 47 | \( 1 - 216 T^{2} + 24532 T^{4} - 1862136 T^{6} + 101777478 T^{8} - 1862136 p^{2} T^{10} + 24532 p^{4} T^{12} - 216 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( 1 + 24 T + 288 T^{2} + 2936 T^{3} + 30556 T^{4} + 284840 T^{5} + 2346080 T^{6} + 19062856 T^{7} + 147591846 T^{8} + 19062856 p T^{9} + 2346080 p^{2} T^{10} + 284840 p^{3} T^{11} + 30556 p^{4} T^{12} + 2936 p^{5} T^{13} + 288 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \) |
| 59 | \( 1 - 22 T + 242 T^{2} - 1870 T^{3} + 8144 T^{4} - 22682 T^{5} + 276606 T^{6} - 4745202 T^{7} + 49894594 T^{8} - 4745202 p T^{9} + 276606 p^{2} T^{10} - 22682 p^{3} T^{11} + 8144 p^{4} T^{12} - 1870 p^{5} T^{13} + 242 p^{6} T^{14} - 22 p^{7} T^{15} + p^{8} T^{16} \) |
| 61 | \( ( 1 - 10 T + 124 T^{2} - 318 T^{3} + 4058 T^{4} - 318 p T^{5} + 124 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 + 6 T + 236 T^{2} + 1014 T^{3} + 22498 T^{4} + 1014 p T^{5} + 236 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( 1 + 10 T + 50 T^{2} + 818 T^{3} + 56 p T^{4} - 33914 T^{5} - 203378 T^{6} - 3779058 T^{7} - 67309230 T^{8} - 3779058 p T^{9} - 203378 p^{2} T^{10} - 33914 p^{3} T^{11} + 56 p^{5} T^{12} + 818 p^{5} T^{13} + 50 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) |
| 73 | \( ( 1 + 2 T + 236 T^{2} + 382 T^{3} + 24538 T^{4} + 382 p T^{5} + 236 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( 1 - 340 T^{2} + 65892 T^{4} - 8434412 T^{6} + 781259126 T^{8} - 8434412 p^{2} T^{10} + 65892 p^{4} T^{12} - 340 p^{6} T^{14} + p^{8} T^{16} \) |
| 83 | \( 1 - 248 T^{2} + 43620 T^{4} - 5170264 T^{6} + 500303558 T^{8} - 5170264 p^{2} T^{10} + 43620 p^{4} T^{12} - 248 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( 1 + 28 T + 392 T^{2} + 4020 T^{3} + 31916 T^{4} + 289180 T^{5} + 3666168 T^{6} + 49450068 T^{7} + 555566950 T^{8} + 49450068 p T^{9} + 3666168 p^{2} T^{10} + 289180 p^{3} T^{11} + 31916 p^{4} T^{12} + 4020 p^{5} T^{13} + 392 p^{6} T^{14} + 28 p^{7} T^{15} + p^{8} T^{16} \) |
| 97 | \( ( 1 - 6 T + 260 T^{2} - 1266 T^{3} + 35326 T^{4} - 1266 p T^{5} + 260 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.21935838656273575774134866803, −6.87648173898165201033931885842, −6.85788341600183512872207704174, −6.78891498862958724633873806002, −6.70901824852370365272094294555, −6.38620527392712089278701628847, −6.13441084724006496807754619305, −5.90696980874424973850839559141, −5.77227257833640586987433775080, −5.59169451660521777453888434258, −5.41496974358630568270895729773, −5.08808158779083335096813891818, −4.79762587947333174023710871602, −4.79025585203389526205423182940, −4.68820107006109882557049776490, −4.58971945404141182116384526896, −4.16516541125219532829994405518, −3.90845422570094098027227911901, −3.87176864324593968150339037411, −3.09654214130448017941086570478, −2.66526470055971884859721712810, −2.49041612812083156781982874075, −1.76599159402419681511857234190, −1.08644087346231578302571292467, −0.891721111062995731338481782431,
0.891721111062995731338481782431, 1.08644087346231578302571292467, 1.76599159402419681511857234190, 2.49041612812083156781982874075, 2.66526470055971884859721712810, 3.09654214130448017941086570478, 3.87176864324593968150339037411, 3.90845422570094098027227911901, 4.16516541125219532829994405518, 4.58971945404141182116384526896, 4.68820107006109882557049776490, 4.79025585203389526205423182940, 4.79762587947333174023710871602, 5.08808158779083335096813891818, 5.41496974358630568270895729773, 5.59169451660521777453888434258, 5.77227257833640586987433775080, 5.90696980874424973850839559141, 6.13441084724006496807754619305, 6.38620527392712089278701628847, 6.70901824852370365272094294555, 6.78891498862958724633873806002, 6.85788341600183512872207704174, 6.87648173898165201033931885842, 7.21935838656273575774134866803
Plot not available for L-functions of degree greater than 10.
