| L(s) = 1 | + 8·2-s + 16·4-s − 53·5-s + 6·7-s − 128·8-s − 424·10-s + 191·11-s − 758·13-s + 48·14-s − 1.02e3·16-s − 340·17-s + 1.76e3·19-s − 848·20-s + 1.52e3·22-s + 3.23e3·23-s + 4.55e3·25-s − 6.06e3·26-s + 96·28-s − 8.91e3·29-s − 1.99e3·31-s − 2.04e3·32-s − 2.72e3·34-s − 318·35-s − 2.05e4·37-s + 1.41e4·38-s + 6.78e3·40-s − 1.76e4·41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1/2·4-s − 0.948·5-s + 0.0462·7-s − 0.707·8-s − 1.34·10-s + 0.475·11-s − 1.24·13-s + 0.0654·14-s − 16-s − 0.285·17-s + 1.12·19-s − 0.474·20-s + 0.673·22-s + 1.27·23-s + 1.45·25-s − 1.75·26-s + 0.0231·28-s − 1.96·29-s − 0.372·31-s − 0.353·32-s − 0.403·34-s − 0.0438·35-s − 2.47·37-s + 1.58·38-s + 0.670·40-s − 1.63·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.538613176\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.538613176\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 - 6 T - 683 p T^{2} - 6 p^{5} T^{3} + p^{10} T^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 53 T - 1743 T^{2} - 89994 T^{3} + 2176954 T^{4} - 89994 p^{5} T^{5} - 1743 p^{10} T^{6} + 53 p^{15} T^{7} + p^{20} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 191 T - 234735 T^{2} + 883566 p T^{3} + 345003244 p^{2} T^{4} + 883566 p^{6} T^{5} - 234735 p^{10} T^{6} - 191 p^{15} T^{7} + p^{20} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 379 T + 488066 T^{2} + 379 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 20 p T - 1792914 T^{2} - 18624000 p T^{3} + 1462296318547 T^{4} - 18624000 p^{6} T^{5} - 1792914 p^{10} T^{6} + 20 p^{16} T^{7} + p^{20} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 1769 T + 1045603 T^{2} + 5074270360 T^{3} - 9537626497976 T^{4} + 5074270360 p^{5} T^{5} + 1045603 p^{10} T^{6} - 1769 p^{15} T^{7} + p^{20} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 3236 T - 4980510 T^{2} - 8347326720 T^{3} + 129944731805059 T^{4} - 8347326720 p^{5} T^{5} - 4980510 p^{10} T^{6} - 3236 p^{15} T^{7} + p^{20} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 4459 T + 37061638 T^{2} + 4459 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 1994 T + 10280849 T^{2} - 126744851310 T^{3} - 893708155041268 T^{4} - 126744851310 p^{5} T^{5} + 10280849 p^{10} T^{6} + 1994 p^{15} T^{7} + p^{20} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 20587 T + 185423563 T^{2} + 2052793425004 T^{3} + 22636782601114654 T^{4} + 2052793425004 p^{5} T^{5} + 185423563 p^{10} T^{6} + 20587 p^{15} T^{7} + p^{20} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 8814 T + 240678562 T^{2} + 8814 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 15853 T + 338678796 T^{2} - 15853 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 33912 T + 462240278 T^{2} - 7769017144224 T^{3} + 156694760249296899 T^{4} - 7769017144224 p^{5} T^{5} + 462240278 p^{10} T^{6} - 33912 p^{15} T^{7} + p^{20} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 49239 T + 1103481011 T^{2} - 23861570178636 T^{3} + 556242294983257398 T^{4} - 23861570178636 p^{5} T^{5} + 1103481011 p^{10} T^{6} - 49239 p^{15} T^{7} + p^{20} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 56735 T + 988331391 T^{2} + 45426593189460 T^{3} + 2162900736067650880 T^{4} + 45426593189460 p^{5} T^{5} + 988331391 p^{10} T^{6} + 56735 p^{15} T^{7} + p^{20} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 67508 T + 1731262802 T^{2} + 76748134547280 T^{3} + 3424209143194800779 T^{4} + 76748134547280 p^{5} T^{5} + 1731262802 p^{10} T^{6} + 67508 p^{15} T^{7} + p^{20} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 75723 T + 1872823325 T^{2} + 87906769364370 T^{3} + 5344056371181586764 T^{4} + 87906769364370 p^{5} T^{5} + 1872823325 p^{10} T^{6} + 75723 p^{15} T^{7} + p^{20} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 8992 T - 681216182 T^{2} - 8992 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 3201 T - 2300766979 T^{2} + 5874250509006 T^{3} + 1021915489991879958 T^{4} + 5874250509006 p^{5} T^{5} - 2300766979 p^{10} T^{6} - 3201 p^{15} T^{7} + p^{20} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 26612 T - 3289450441 T^{2} - 57387814991556 T^{3} + 4333754463066999968 T^{4} - 57387814991556 p^{5} T^{5} - 3289450441 p^{10} T^{6} + 26612 p^{15} T^{7} + p^{20} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 949 T + 367057696 T^{2} - 949 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 176562 T + 12318337010 T^{2} - 1357352851108032 T^{3} + \)\(15\!\cdots\!99\)\( T^{4} - 1357352851108032 p^{5} T^{5} + 12318337010 p^{10} T^{6} - 176562 p^{15} T^{7} + p^{20} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 129423 T + 11942811256 T^{2} + 129423 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.069879192122574898495856516542, −8.740659253089825151546420509926, −8.370005846194992190988347012092, −7.65066742977170227177663064754, −7.54755158398408503166965863555, −7.54417407631467050454184774319, −7.06833211016014294132596465115, −6.76495015722998805396506992118, −6.73206824287478074101875780849, −5.81834115349266830926858240154, −5.76361348712872961936788925217, −5.59202045253676122459458449674, −5.02111176747107316413682331182, −4.91503309142647855348890016267, −4.59976368735826400567093840977, −4.12457067199298598192794404726, −3.76025057644362189423688096054, −3.74275675750012345361824928925, −3.07636583610418081012046819235, −2.79605513249982321686827325637, −2.58068510479999918438822279089, −1.68054151491906424939103888895, −1.46635481624463237676080239482, −0.61348372369356748237067130245, −0.32654658014322787206962656965,
0.32654658014322787206962656965, 0.61348372369356748237067130245, 1.46635481624463237676080239482, 1.68054151491906424939103888895, 2.58068510479999918438822279089, 2.79605513249982321686827325637, 3.07636583610418081012046819235, 3.74275675750012345361824928925, 3.76025057644362189423688096054, 4.12457067199298598192794404726, 4.59976368735826400567093840977, 4.91503309142647855348890016267, 5.02111176747107316413682331182, 5.59202045253676122459458449674, 5.76361348712872961936788925217, 5.81834115349266830926858240154, 6.73206824287478074101875780849, 6.76495015722998805396506992118, 7.06833211016014294132596465115, 7.54417407631467050454184774319, 7.54755158398408503166965863555, 7.65066742977170227177663064754, 8.370005846194992190988347012092, 8.740659253089825151546420509926, 9.069879192122574898495856516542
