L(s) = 1 | − 2-s − 4-s − 5-s + 3·8-s + 9-s + 10-s − 8·11-s − 4·13-s − 16-s + 4·17-s − 18-s + 8·19-s + 20-s + 8·22-s + 8·23-s − 2·25-s + 4·26-s + 4·29-s − 8·31-s − 5·32-s − 4·34-s − 36-s − 4·37-s − 8·38-s − 3·40-s + 4·41-s + 8·44-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.447·5-s + 1.06·8-s + 1/3·9-s + 0.316·10-s − 2.41·11-s − 1.10·13-s − 1/4·16-s + 0.970·17-s − 0.235·18-s + 1.83·19-s + 0.223·20-s + 1.70·22-s + 1.66·23-s − 2/5·25-s + 0.784·26-s + 0.742·29-s − 1.43·31-s − 0.883·32-s − 0.685·34-s − 1/6·36-s − 0.657·37-s − 1.29·38-s − 0.474·40-s + 0.624·41-s + 1.20·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2951333844\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2951333844\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 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$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−19.8937036145, −19.1104624590, −19.0038467807, −18.1058537571, −18.0808264901, −17.2359534775, −16.5594222959, −15.9590260302, −15.5766594288, −14.7395675207, −14.0069258505, −13.4092975284, −12.6461787614, −12.3221586340, −11.0205906871, −10.6789224512, −9.65679775515, −9.52345167581, −8.17328234983, −7.66488013442, −7.26958585489, −5.23920392625, −5.01357758336, −3.14826068230,
3.14826068230, 5.01357758336, 5.23920392625, 7.26958585489, 7.66488013442, 8.17328234983, 9.52345167581, 9.65679775515, 10.6789224512, 11.0205906871, 12.3221586340, 12.6461787614, 13.4092975284, 14.0069258505, 14.7395675207, 15.5766594288, 15.9590260302, 16.5594222959, 17.2359534775, 18.0808264901, 18.1058537571, 19.0038467807, 19.1104624590, 19.8937036145