| L(s) = 1 | − 2-s + 3-s + 4-s + 3·5-s − 6-s − 7-s − 3·8-s − 9-s − 3·10-s + 12-s + 3·13-s + 14-s + 3·15-s + 16-s − 7·17-s + 18-s + 4·19-s + 3·20-s − 21-s − 3·24-s + 25-s − 3·26-s − 28-s − 3·30-s − 6·31-s + 32-s + 7·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.34·5-s − 0.408·6-s − 0.377·7-s − 1.06·8-s − 1/3·9-s − 0.948·10-s + 0.288·12-s + 0.832·13-s + 0.267·14-s + 0.774·15-s + 1/4·16-s − 1.69·17-s + 0.235·18-s + 0.917·19-s + 0.670·20-s − 0.218·21-s − 0.612·24-s + 1/5·25-s − 0.588·26-s − 0.188·28-s − 0.547·30-s − 1.07·31-s + 0.176·32-s + 1.20·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5803 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5803 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8209624933\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8209624933\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| 829 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 34 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 7 T + 32 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $D_{4}$ | \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_4$ | \( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 6 T + 62 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 59 | $D_{4}$ | \( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 15 T + 168 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 22 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $D_{4}$ | \( 1 + 4 T - 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 9 T + 44 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
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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
−17.5553624912, −16.8758159513, −16.3421578819, −15.6872786269, −15.5100776913, −14.6937663431, −14.2765019911, −13.5224128746, −13.3843596070, −12.8564309906, −11.8561610036, −11.4893303537, −10.8440976560, −10.1364502870, −9.62261865919, −9.14567121747, −8.66331539655, −8.27160921510, −6.98017718586, −6.71727717864, −5.85452333348, −5.36528334321, −3.90886059645, −2.84686690829, −2.02790848567,
2.02790848567, 2.84686690829, 3.90886059645, 5.36528334321, 5.85452333348, 6.71727717864, 6.98017718586, 8.27160921510, 8.66331539655, 9.14567121747, 9.62261865919, 10.1364502870, 10.8440976560, 11.4893303537, 11.8561610036, 12.8564309906, 13.3843596070, 13.5224128746, 14.2765019911, 14.6937663431, 15.5100776913, 15.6872786269, 16.3421578819, 16.8758159513, 17.5553624912
