| L(s) = 1 | + 2-s − 4·3-s + 4-s − 2·5-s − 4·6-s − 3·7-s + 8-s + 7·9-s − 2·10-s − 5·11-s − 4·12-s − 13-s − 3·14-s + 8·15-s + 16-s + 4·17-s + 7·18-s − 2·19-s − 2·20-s + 12·21-s − 5·22-s − 4·24-s + 25-s − 26-s − 4·27-s − 3·28-s − 4·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 2.30·3-s + 1/2·4-s − 0.894·5-s − 1.63·6-s − 1.13·7-s + 0.353·8-s + 7/3·9-s − 0.632·10-s − 1.50·11-s − 1.15·12-s − 0.277·13-s − 0.801·14-s + 2.06·15-s + 1/4·16-s + 0.970·17-s + 1.64·18-s − 0.458·19-s − 0.447·20-s + 2.61·21-s − 1.06·22-s − 0.816·24-s + 1/5·25-s − 0.196·26-s − 0.769·27-s − 0.566·28-s − 0.742·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 683 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 25 T + p T^{2} ) \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $D_{4}$ | \( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 12 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 5 T + 21 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + T - 5 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + p T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T - 16 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 49 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 3 T + 15 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 2 T - 11 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - T - p T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - T + 66 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2 T + 49 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 21 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 5 T + 22 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 129 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 114 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 89 | $D_{4}$ | \( 1 - 7 T + 186 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - T + 113 T^{2} - p T^{3} + p^{2} T^{4} \) |
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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
−17.5041936793, −16.9216962949, −16.5241439747, −16.2705024489, −15.6364659122, −15.4878524404, −14.6712752832, −13.9855295102, −13.0790787560, −12.7635140808, −12.4649507010, −11.7340532669, −11.5351562227, −10.8755189361, −10.3447684594, −10.0465116886, −8.95914764098, −7.79264697180, −7.47384296855, −6.49170642107, −6.13999687052, −5.32572325793, −5.06120187130, −3.97703351627, −2.95625603827, 0,
2.95625603827, 3.97703351627, 5.06120187130, 5.32572325793, 6.13999687052, 6.49170642107, 7.47384296855, 7.79264697180, 8.95914764098, 10.0465116886, 10.3447684594, 10.8755189361, 11.5351562227, 11.7340532669, 12.4649507010, 12.7635140808, 13.0790787560, 13.9855295102, 14.6712752832, 15.4878524404, 15.6364659122, 16.2705024489, 16.5241439747, 16.9216962949, 17.5041936793
