| L(s) = 1 | − 4·2-s − 4·3-s + 8·4-s + 16·6-s − 4·7-s − 12·8-s + 4·9-s − 4·11-s − 32·12-s + 16·14-s + 15·16-s − 12·17-s − 16·18-s − 8·19-s + 16·21-s + 16·22-s − 12·23-s + 48·24-s + 4·27-s − 32·28-s + 4·29-s − 12·31-s − 16·32-s + 16·33-s + 48·34-s + 32·36-s + 20·37-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 2.30·3-s + 4·4-s + 6.53·6-s − 1.51·7-s − 4.24·8-s + 4/3·9-s − 1.20·11-s − 9.23·12-s + 4.27·14-s + 15/4·16-s − 2.91·17-s − 3.77·18-s − 1.83·19-s + 3.49·21-s + 3.41·22-s − 2.50·23-s + 9.79·24-s + 0.769·27-s − 6.04·28-s + 0.742·29-s − 2.15·31-s − 2.82·32-s + 2.78·33-s + 8.23·34-s + 16/3·36-s + 3.28·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| good | 2 | $D_4\times C_2$ | \( 1 + p^{2} T + p^{3} T^{2} + 3 p^{2} T^{3} + 17 T^{4} + 3 p^{3} T^{5} + p^{5} T^{6} + p^{5} T^{7} + p^{4} T^{8} \) |
| 3 | $D_4\times C_2$ | \( 1 + 4 T + 4 p T^{2} + 28 T^{3} + 56 T^{4} + 28 p T^{5} + 4 p^{3} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 + 4 T + 12 T^{2} + 44 T^{3} + 120 T^{4} + 44 p T^{5} + 12 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 4 T + 12 T^{2} + 60 T^{3} + 184 T^{4} + 60 p T^{5} + 12 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + 24 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 184 T^{3} + 1042 T^{4} + 184 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 12 T + 68 T^{2} + 316 T^{3} + 1496 T^{4} + 316 p T^{5} + 68 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 4 T + 22 T^{2} - 244 T^{3} + 930 T^{4} - 244 p T^{5} + 22 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 12 T + 108 T^{2} + 804 T^{3} + 5112 T^{4} + 804 p T^{5} + 108 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 20 T + 150 T^{2} - 500 T^{3} + 1250 T^{4} - 500 p T^{5} + 150 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2$$\times$$C_2^2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} ) \) |
| 43 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 376 T^{3} + 4402 T^{4} + 376 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 20 T + 200 T^{2} - 20 p T^{3} + p^{2} T^{4} )( 1 + 20 T + 200 T^{2} + 20 p T^{3} + p^{2} T^{4} ) \) |
| 61 | $D_4\times C_2$ | \( 1 + 50 T^{2} - 720 T^{3} + 1250 T^{4} - 720 p T^{5} + 50 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 220 T^{2} + 20566 T^{4} - 220 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 20 T + 100 T^{2} + 20 p T^{3} - 23400 T^{4} + 20 p^{2} T^{5} + 100 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 + 98 T^{2} + 672 T^{3} + 4802 T^{4} + 672 p T^{5} + 98 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 4 T + 12 T^{2} + 836 T^{3} - 3400 T^{4} + 836 p T^{5} + 12 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 16 T + 128 T^{2} + 1264 T^{3} + 12466 T^{4} + 1264 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 224 T^{2} + 27874 T^{4} - 224 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 4 T + 54 T^{2} + 1388 T^{3} - 4894 T^{4} + 1388 p T^{5} + 54 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.461712094088775646650720759822, −8.395170124393985841844533636386, −8.247876836264968698360555225332, −8.043099558011723850337980177806, −7.84276778428535037952558298962, −7.41365566151151161894465217570, −6.83732561576051257122083323241, −6.74139466625076497267501010283, −6.72645989616945193200399518182, −6.51525938650821944907948057307, −6.25003174254374457773934081906, −5.83150170105886121986071238743, −5.78602528569735038141163518109, −5.74969100367221134875452148429, −4.96999559640665771319244717240, −4.95124961730577736204645377171, −4.62157351961607601781014946113, −4.05241045658196203350523733790, −3.99425172762489601557091065438, −3.42004887698769473364810613663, −3.01217169934508413766871457344, −2.39289955090906246927740323385, −2.35485851093939593475873485339, −2.01830063658283705206385857218, −1.33682807989344830247393465810, 0, 0, 0, 0,
1.33682807989344830247393465810, 2.01830063658283705206385857218, 2.35485851093939593475873485339, 2.39289955090906246927740323385, 3.01217169934508413766871457344, 3.42004887698769473364810613663, 3.99425172762489601557091065438, 4.05241045658196203350523733790, 4.62157351961607601781014946113, 4.95124961730577736204645377171, 4.96999559640665771319244717240, 5.74969100367221134875452148429, 5.78602528569735038141163518109, 5.83150170105886121986071238743, 6.25003174254374457773934081906, 6.51525938650821944907948057307, 6.72645989616945193200399518182, 6.74139466625076497267501010283, 6.83732561576051257122083323241, 7.41365566151151161894465217570, 7.84276778428535037952558298962, 8.043099558011723850337980177806, 8.247876836264968698360555225332, 8.395170124393985841844533636386, 8.461712094088775646650720759822
