| L(s) = 1 | − 2-s − 2·5-s + 8-s − 9-s + 2·10-s − 13-s − 16-s + 17-s + 18-s + 25-s + 26-s + 29-s − 34-s + 37-s − 2·40-s + 41-s + 2·45-s − 49-s − 50-s − 2·53-s − 58-s + 61-s + 64-s + 2·65-s − 72-s − 2·73-s − 74-s + ⋯ |
| L(s) = 1 | − 2-s − 2·5-s + 8-s − 9-s + 2·10-s − 13-s − 16-s + 17-s + 18-s + 25-s + 26-s + 29-s − 34-s + 37-s − 2·40-s + 41-s + 2·45-s − 49-s − 50-s − 2·53-s − 58-s + 61-s + 64-s + 2·65-s − 72-s − 2·73-s − 74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1055166369\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1055166369\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 13 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.87294873793952215742661356483, −15.86901024324633578237109774276, −14.96124058887919042375486728325, −14.36669503053971576318634699798, −14.12767115902838684115562210190, −13.06765961975395203124255805669, −12.40865268975567652680377933239, −11.93591325352333454299037842375, −11.24632239324280198745131969122, −11.08546064930377023602876475562, −9.896784789924800205671186547144, −9.709118972498971638443287007627, −8.532649398844231199448686039577, −8.339948637340426939401113737946, −7.53136053749764317001096925301, −7.39766540537995423654738286953, −6.04668941269644592254494882943, −4.87537373101459305382914912093, −4.16168360291039984927090654039, −3.05212385545771777947032732532,
3.05212385545771777947032732532, 4.16168360291039984927090654039, 4.87537373101459305382914912093, 6.04668941269644592254494882943, 7.39766540537995423654738286953, 7.53136053749764317001096925301, 8.339948637340426939401113737946, 8.532649398844231199448686039577, 9.709118972498971638443287007627, 9.896784789924800205671186547144, 11.08546064930377023602876475562, 11.24632239324280198745131969122, 11.93591325352333454299037842375, 12.40865268975567652680377933239, 13.06765961975395203124255805669, 14.12767115902838684115562210190, 14.36669503053971576318634699798, 14.96124058887919042375486728325, 15.86901024324633578237109774276, 15.87294873793952215742661356483
