| L(s) = 1 | − 4·2-s + 4·4-s + 16·8-s − 18·9-s − 64·16-s + 72·18-s + 48·25-s − 160·29-s + 64·32-s − 72·36-s + 48·37-s − 192·50-s + 180·53-s + 640·58-s + 192·64-s − 288·72-s − 192·74-s + 81·81-s + 192·100-s − 720·106-s − 240·109-s + 120·113-s − 640·116-s − 242·121-s + 127-s − 768·128-s + 131-s + ⋯ |
| L(s) = 1 | − 2·2-s + 4-s + 2·8-s − 2·9-s − 4·16-s + 4·18-s + 1.91·25-s − 5.51·29-s + 2·32-s − 2·36-s + 1.29·37-s − 3.83·50-s + 3.39·53-s + 11.0·58-s + 3·64-s − 4·72-s − 2.59·74-s + 81-s + 1.91·100-s − 6.79·106-s − 2.20·109-s + 1.06·113-s − 5.51·116-s − 2·121-s + 0.00787·127-s − 6·128-s + 0.00763·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.02792148789\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02792148789\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 + p T + p^{2} T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 5 | $C_2^3$ | \( 1 - 48 T^{2} + 1679 T^{4} - 48 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 240 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 + 480 T^{2} + 146879 T^{4} + 480 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 40 T + p^{2} T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 24 T - 793 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 1440 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 90 T + 5291 T^{2} - 90 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 61 | $C_2^3$ | \( 1 + 2640 T^{2} - 6876241 T^{4} + 2640 p^{4} T^{6} + p^{8} T^{8} \) |
| 67 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 73 | $C_2^3$ | \( 1 + 10560 T^{2} + 83115359 T^{4} + 10560 p^{4} T^{6} + p^{8} T^{8} \) |
| 79 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 89 | $C_2^3$ | \( 1 - 12480 T^{2} + 93008159 T^{4} - 12480 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 - 18720 T^{2} + p^{4} T^{4} )^{2} \) |
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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
−8.883253036949959740882209147274, −8.837682638665124225857892499133, −8.402074294298609866684959566947, −8.321798879723122959348878690379, −7.891416830075019561770404743716, −7.62073644805965420570409585220, −7.50768944249032107557798457993, −7.11832383828138599828419622078, −7.02134768434682509181185119346, −6.58603455138705337937583758316, −6.10445916204506486360124568036, −5.66039179590342428210204253687, −5.49780886910951854378294515991, −5.49012020515541715314959942337, −4.87707834713967950220343730351, −4.68933470524542603314913856956, −3.91100139739326081580053935887, −3.88783155843764733070759406546, −3.63613174634580629027683636566, −2.86684889731592295233855442287, −2.44409237265019865090841488400, −2.14844261708241882081039850007, −1.47879645276394317695431017811, −0.925135470212029462559113337022, −0.092081982490783129469417749075,
0.092081982490783129469417749075, 0.925135470212029462559113337022, 1.47879645276394317695431017811, 2.14844261708241882081039850007, 2.44409237265019865090841488400, 2.86684889731592295233855442287, 3.63613174634580629027683636566, 3.88783155843764733070759406546, 3.91100139739326081580053935887, 4.68933470524542603314913856956, 4.87707834713967950220343730351, 5.49012020515541715314959942337, 5.49780886910951854378294515991, 5.66039179590342428210204253687, 6.10445916204506486360124568036, 6.58603455138705337937583758316, 7.02134768434682509181185119346, 7.11832383828138599828419622078, 7.50768944249032107557798457993, 7.62073644805965420570409585220, 7.891416830075019561770404743716, 8.321798879723122959348878690379, 8.402074294298609866684959566947, 8.837682638665124225857892499133, 8.883253036949959740882209147274
