| L(s) = 1 | + 3-s − 4-s − 5-s − 7-s + 2·8-s − 2·9-s − 2·11-s − 12-s − 2·13-s − 15-s + 16-s + 3·17-s − 7·19-s + 20-s − 21-s + 10·23-s + 2·24-s + 25-s − 2·27-s + 28-s − 4·29-s + 31-s − 4·32-s − 2·33-s + 35-s + 2·36-s + 37-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1/2·4-s − 0.447·5-s − 0.377·7-s + 0.707·8-s − 2/3·9-s − 0.603·11-s − 0.288·12-s − 0.554·13-s − 0.258·15-s + 1/4·16-s + 0.727·17-s − 1.60·19-s + 0.223·20-s − 0.218·21-s + 2.08·23-s + 0.408·24-s + 1/5·25-s − 0.384·27-s + 0.188·28-s − 0.742·29-s + 0.179·31-s − 0.707·32-s − 0.348·33-s + 0.169·35-s + 1/3·36-s + 0.164·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2154 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2154 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6429869962\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6429869962\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_2$ | \( ( 1 - T )( 1 + T + p T^{2} ) \) |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| 359 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 24 T + p T^{2} ) \) |
| good | 5 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $D_{4}$ | \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 3 T + 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 7 T + 42 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $D_{4}$ | \( 1 - T + 30 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - T + 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $D_{4}$ | \( 1 + 5 T + 74 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 61 | $D_{4}$ | \( 1 - 6 T - 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $D_{4}$ | \( 1 - 3 T + 12 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2 T - 98 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 11 T + 130 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−18.8805814711, −18.4802903015, −17.5120009803, −17.0438118657, −16.7682728802, −16.1309192780, −15.2707037944, −14.8940150380, −14.5016281311, −13.7750958827, −13.1964429057, −12.7720736619, −12.2174719335, −11.2120776809, −10.7997260850, −10.1146510525, −9.32777073683, −8.71719068195, −8.15240908043, −7.45944563647, −6.72255606834, −5.51624256743, −4.78852528928, −3.73478924331, −2.66706983613,
2.66706983613, 3.73478924331, 4.78852528928, 5.51624256743, 6.72255606834, 7.45944563647, 8.15240908043, 8.71719068195, 9.32777073683, 10.1146510525, 10.7997260850, 11.2120776809, 12.2174719335, 12.7720736619, 13.1964429057, 13.7750958827, 14.5016281311, 14.8940150380, 15.2707037944, 16.1309192780, 16.7682728802, 17.0438118657, 17.5120009803, 18.4802903015, 18.8805814711
