| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s + 7-s − 3·8-s − 2·9-s − 10-s − 11-s + 12-s − 3·13-s − 14-s + 15-s + 16-s + 2·17-s + 2·18-s − 19-s + 20-s + 21-s + 22-s − 3·23-s − 3·24-s + 5·25-s + 3·26-s − 2·27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.377·7-s − 1.06·8-s − 2/3·9-s − 0.316·10-s − 0.301·11-s + 0.288·12-s − 0.832·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.485·17-s + 0.471·18-s − 0.229·19-s + 0.223·20-s + 0.218·21-s + 0.213·22-s − 0.625·23-s − 0.612·24-s + 25-s + 0.588·26-s − 0.384·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3003 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3003 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6267721971\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6267721971\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + T + 10 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 7 T + 36 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 9 T + 78 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 6 T + 10 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 + 3 T - 6 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 3 T - 26 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $D_{4}$ | \( 1 + 21 T + 252 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - T + 54 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 5 T + 66 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - T - 96 T^{2} - p T^{3} + p^{2} T^{4} \) |
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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
−18.1922733333, −17.6902147432, −17.3670963399, −16.7311999135, −16.2882659864, −15.5556777118, −14.9627374799, −14.5445764997, −14.2315582053, −13.3536007582, −12.8890931025, −12.0362812872, −11.6274379360, −11.0407886864, −10.1258592420, −9.77705231696, −9.07584568004, −8.47442266599, −8.01437802908, −7.18935141367, −6.33012834057, −5.62837115223, −4.64525819134, −3.13204824911, −2.32829175218,
2.32829175218, 3.13204824911, 4.64525819134, 5.62837115223, 6.33012834057, 7.18935141367, 8.01437802908, 8.47442266599, 9.07584568004, 9.77705231696, 10.1258592420, 11.0407886864, 11.6274379360, 12.0362812872, 12.8890931025, 13.3536007582, 14.2315582053, 14.5445764997, 14.9627374799, 15.5556777118, 16.2882659864, 16.7311999135, 17.3670963399, 17.6902147432, 18.1922733333
