| L(s) = 1 | − 2·5-s + 2·11-s − 4·13-s − 6·17-s − 4·19-s + 6·23-s + 5·25-s − 4·31-s − 10·37-s + 4·41-s − 8·43-s − 4·47-s − 12·53-s − 4·55-s − 12·59-s + 6·61-s + 8·65-s + 4·67-s + 28·71-s − 2·73-s + 8·79-s − 32·83-s + 12·85-s + 6·89-s + 8·95-s + 36·97-s − 14·101-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.603·11-s − 1.10·13-s − 1.45·17-s − 0.917·19-s + 1.25·23-s + 25-s − 0.718·31-s − 1.64·37-s + 0.624·41-s − 1.21·43-s − 0.583·47-s − 1.64·53-s − 0.539·55-s − 1.56·59-s + 0.768·61-s + 0.992·65-s + 0.488·67-s + 3.32·71-s − 0.234·73-s + 0.900·79-s − 3.51·83-s + 1.30·85-s + 0.635·89-s + 0.820·95-s + 3.65·97-s − 1.39·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3090695749\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3090695749\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 4 T - 31 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
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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
−8.892532111206215314081814045533, −8.259823584698242129600574003328, −8.158978306695114281518236043958, −7.55504578972573212675448590662, −7.33250340666059212817064818339, −6.75487803874930213311765613616, −6.61089258524233622314130760797, −6.49051977246742497052624035502, −5.72059138298391011076206441038, −5.12040408277451557448104379783, −4.93856985383457531089945309857, −4.59953846223430251767407471453, −4.19561667137027706637369340172, −3.56393132338580887204049919605, −3.43227728945779337899918337910, −2.78608199452191970089853165671, −2.24915439229303985777566669068, −1.83716675895772120153222614565, −1.13150254014342935058229240718, −0.17587506248643707713497789691,
0.17587506248643707713497789691, 1.13150254014342935058229240718, 1.83716675895772120153222614565, 2.24915439229303985777566669068, 2.78608199452191970089853165671, 3.43227728945779337899918337910, 3.56393132338580887204049919605, 4.19561667137027706637369340172, 4.59953846223430251767407471453, 4.93856985383457531089945309857, 5.12040408277451557448104379783, 5.72059138298391011076206441038, 6.49051977246742497052624035502, 6.61089258524233622314130760797, 6.75487803874930213311765613616, 7.33250340666059212817064818339, 7.55504578972573212675448590662, 8.158978306695114281518236043958, 8.259823584698242129600574003328, 8.892532111206215314081814045533
