L(s) = 1 | − 2·3-s − 4-s − 5-s − 7-s + 2·8-s + 3·9-s − 8·11-s + 2·12-s + 4·13-s + 2·15-s + 16-s + 4·17-s + 20-s + 2·21-s + 8·23-s − 4·24-s − 2·25-s − 4·27-s + 28-s − 4·29-s − 4·32-s + 16·33-s + 35-s − 3·36-s − 20·37-s − 8·39-s − 2·40-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 1/2·4-s − 0.447·5-s − 0.377·7-s + 0.707·8-s + 9-s − 2.41·11-s + 0.577·12-s + 1.10·13-s + 0.516·15-s + 1/4·16-s + 0.970·17-s + 0.223·20-s + 0.436·21-s + 1.66·23-s − 0.816·24-s − 2/5·25-s − 0.769·27-s + 0.188·28-s − 0.742·29-s − 0.707·32-s + 2.78·33-s + 0.169·35-s − 1/2·36-s − 3.28·37-s − 1.28·39-s − 0.316·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3042326426\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3042326426\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
good | 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−19.2406986299, −19.1104624590, −18.4485210975, −18.1058537571, −17.2359534775, −16.8741179353, −15.9590260302, −15.8196150869, −15.2472562838, −14.0069258505, −13.4821859369, −12.6521323881, −12.6461787614, −11.3731892768, −10.6789224512, −10.5195197796, −9.52345167581, −8.35691896663, −7.66488013442, −6.84552480603, −5.33985014788, −5.23920392625, −3.62482887082,
3.62482887082, 5.23920392625, 5.33985014788, 6.84552480603, 7.66488013442, 8.35691896663, 9.52345167581, 10.5195197796, 10.6789224512, 11.3731892768, 12.6461787614, 12.6521323881, 13.4821859369, 14.0069258505, 15.2472562838, 15.8196150869, 15.9590260302, 16.8741179353, 17.2359534775, 18.1058537571, 18.4485210975, 19.1104624590, 19.2406986299