| L(s) = 1 | − 2-s + 3·3-s + 4-s + 5-s − 3·6-s − 2·7-s − 3·8-s + 3·9-s − 10-s − 11-s + 3·12-s − 13-s + 2·14-s + 3·15-s + 16-s − 2·17-s − 3·18-s + 2·19-s + 20-s − 6·21-s + 22-s − 23-s − 9·24-s + 5·25-s + 26-s − 2·28-s − 4·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.73·3-s + 1/2·4-s + 0.447·5-s − 1.22·6-s − 0.755·7-s − 1.06·8-s + 9-s − 0.316·10-s − 0.301·11-s + 0.866·12-s − 0.277·13-s + 0.534·14-s + 0.774·15-s + 1/4·16-s − 0.485·17-s − 0.707·18-s + 0.458·19-s + 0.223·20-s − 1.30·21-s + 0.213·22-s − 0.208·23-s − 1.83·24-s + 25-s + 0.196·26-s − 0.377·28-s − 0.742·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5863 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5863 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9611301282\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9611301282\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - T - 26 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 7 T + 16 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + T + 18 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 12 T + 126 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 7 T + 86 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 3 T + 26 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 4 T + 110 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 89 | $D_{4}$ | \( 1 + 5 T + 60 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 3 T - 72 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
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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
−17.3297643335, −16.7838127833, −16.1826064187, −15.7879736874, −15.1337109528, −14.7607498894, −14.4017544081, −13.6470051121, −13.3171232072, −12.7909361971, −12.1124898953, −11.4139043860, −10.8085555333, −9.89034711083, −9.60543855845, −9.18736661168, −8.47511352577, −8.27580024390, −7.34937010141, −6.70692209701, −6.04808015393, −5.00744937277, −3.65228580439, −2.89109700058, −2.31460730519,
2.31460730519, 2.89109700058, 3.65228580439, 5.00744937277, 6.04808015393, 6.70692209701, 7.34937010141, 8.27580024390, 8.47511352577, 9.18736661168, 9.60543855845, 9.89034711083, 10.8085555333, 11.4139043860, 12.1124898953, 12.7909361971, 13.3171232072, 13.6470051121, 14.4017544081, 14.7607498894, 15.1337109528, 15.7879736874, 16.1826064187, 16.7838127833, 17.3297643335
