| L(s) = 1 | − 2-s + 2·3-s − 4-s − 4·5-s − 2·6-s − 7-s + 3·8-s + 3·9-s + 4·10-s − 2·12-s − 4·13-s + 14-s − 8·15-s − 16-s − 4·17-s − 3·18-s + 8·19-s + 4·20-s − 2·21-s + 8·23-s + 6·24-s + 2·25-s + 4·26-s + 4·27-s + 28-s + 4·29-s + 8·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s − 1/2·4-s − 1.78·5-s − 0.816·6-s − 0.377·7-s + 1.06·8-s + 9-s + 1.26·10-s − 0.577·12-s − 1.10·13-s + 0.267·14-s − 2.06·15-s − 1/4·16-s − 0.970·17-s − 0.707·18-s + 1.83·19-s + 0.894·20-s − 0.436·21-s + 1.66·23-s + 1.22·24-s + 2/5·25-s + 0.784·26-s + 0.769·27-s + 0.188·28-s + 0.742·29-s + 1.46·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3802339773\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3802339773\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−19.8937036145, −19.6678488176, −19.0038467807, −18.4192282680, −18.0808264901, −17.1598241820, −16.5594222959, −15.8734598001, −15.5766594288, −14.7395675207, −14.5135323404, −13.4092975284, −13.1937925053, −12.3221586340, −11.5559508175, −11.0205906871, −9.72466121073, −9.65679775515, −8.67236519668, −8.17328234983, −7.26958585489, −7.25047783803, −5.01357758336, −4.13559084051, −3.14826068230,
3.14826068230, 4.13559084051, 5.01357758336, 7.25047783803, 7.26958585489, 8.17328234983, 8.67236519668, 9.65679775515, 9.72466121073, 11.0205906871, 11.5559508175, 12.3221586340, 13.1937925053, 13.4092975284, 14.5135323404, 14.7395675207, 15.5766594288, 15.8734598001, 16.5594222959, 17.1598241820, 18.0808264901, 18.4192282680, 19.0038467807, 19.6678488176, 19.8937036145
