| L(s) = 1 | − 2-s + 4-s − 8-s + 4·9-s + 6·13-s + 16-s + 2·17-s − 4·18-s − 10·19-s + 4·25-s − 6·26-s − 32-s − 2·34-s + 4·36-s + 10·38-s + 4·43-s − 4·47-s + 49-s − 4·50-s + 6·52-s − 4·53-s + 10·59-s + 64-s + 8·67-s + 2·68-s − 4·72-s − 10·76-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 4/3·9-s + 1.66·13-s + 1/4·16-s + 0.485·17-s − 0.942·18-s − 2.29·19-s + 4/5·25-s − 1.17·26-s − 0.176·32-s − 0.342·34-s + 2/3·36-s + 1.62·38-s + 0.609·43-s − 0.583·47-s + 1/7·49-s − 0.565·50-s + 0.832·52-s − 0.549·53-s + 1.30·59-s + 1/8·64-s + 0.977·67-s + 0.242·68-s − 0.471·72-s − 1.14·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 226576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 226576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.476266289\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.476266289\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 17 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
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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
−8.884581525566642698416897935029, −8.676699125510827134172190531408, −7.986630771286866966369496215105, −7.85676673336187890492122820765, −7.01006714696061990073040860193, −6.63290902567677530960988771730, −6.35726803455625258581806981979, −5.76890300458216837594230191934, −5.03032494260385479099444988530, −4.36253091468286410323631401236, −3.90735517211762648242317692959, −3.36630814048531472613877922053, −2.36379285431485949691075843228, −1.70845584969769242768202823457, −0.907847837386124395986526381884,
0.907847837386124395986526381884, 1.70845584969769242768202823457, 2.36379285431485949691075843228, 3.36630814048531472613877922053, 3.90735517211762648242317692959, 4.36253091468286410323631401236, 5.03032494260385479099444988530, 5.76890300458216837594230191934, 6.35726803455625258581806981979, 6.63290902567677530960988771730, 7.01006714696061990073040860193, 7.85676673336187890492122820765, 7.986630771286866966369496215105, 8.676699125510827134172190531408, 8.884581525566642698416897935029
