| L(s) = 1 | − 2·2-s − 3·3-s + 6·6-s + 3·7-s + 4·8-s + 4·9-s − 13-s − 6·14-s − 4·16-s − 6·17-s − 8·18-s + 19-s − 9·21-s − 7·23-s − 12·24-s − 4·25-s + 2·26-s − 6·27-s + 2·29-s + 6·31-s + 12·34-s − 5·37-s − 2·38-s + 3·39-s − 3·41-s + 18·42-s − 9·43-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.73·3-s + 2.44·6-s + 1.13·7-s + 1.41·8-s + 4/3·9-s − 0.277·13-s − 1.60·14-s − 16-s − 1.45·17-s − 1.88·18-s + 0.229·19-s − 1.96·21-s − 1.45·23-s − 2.44·24-s − 4/5·25-s + 0.392·26-s − 1.15·27-s + 0.371·29-s + 1.07·31-s + 2.05·34-s − 0.821·37-s − 0.324·38-s + 0.480·39-s − 0.468·41-s + 2.77·42-s − 1.37·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5867 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5867 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5867 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 56 T + p T^{2} ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $C_4$ | \( 1 + p T + 5 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 3 T + 12 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $D_{4}$ | \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $D_{4}$ | \( 1 - 2 T + 26 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 6 T + 48 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 5 T + 61 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 3 T + 37 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 9 T + 67 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + T - 32 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 6 T + 92 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 19 T + 198 T^{2} - 19 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 5 T + 46 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 3 T - 36 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 18 T + 212 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 11 T + 93 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 150 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 3 T - 56 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
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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
−17.6312518964, −17.2655612410, −16.9439581186, −16.3329756895, −15.6549888372, −15.4193859665, −14.3784080925, −14.0042047954, −13.3386294344, −12.8276577102, −11.7876149559, −11.7018180282, −11.2388070593, −10.5559546139, −9.93349085014, −9.74904125220, −8.71947414133, −8.27432257566, −7.83857909408, −6.85687468611, −6.25499139779, −5.31851034676, −4.82588421631, −4.11851280697, −1.77255427529, 0,
1.77255427529, 4.11851280697, 4.82588421631, 5.31851034676, 6.25499139779, 6.85687468611, 7.83857909408, 8.27432257566, 8.71947414133, 9.74904125220, 9.93349085014, 10.5559546139, 11.2388070593, 11.7018180282, 11.7876149559, 12.8276577102, 13.3386294344, 14.0042047954, 14.3784080925, 15.4193859665, 15.6549888372, 16.3329756895, 16.9439581186, 17.2655612410, 17.6312518964
