| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 5·7-s + 8-s − 10-s − 11-s − 12-s + 2·13-s + 5·14-s − 15-s − 3·16-s − 9·17-s + 4·19-s + 20-s + 5·21-s + 22-s − 24-s + 6·25-s − 2·26-s − 2·27-s − 5·28-s + 7·29-s + 30-s + 31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 1.88·7-s + 0.353·8-s − 0.316·10-s − 0.301·11-s − 0.288·12-s + 0.554·13-s + 1.33·14-s − 0.258·15-s − 3/4·16-s − 2.18·17-s + 0.917·19-s + 0.223·20-s + 1.09·21-s + 0.213·22-s − 0.204·24-s + 6/5·25-s − 0.392·26-s − 0.384·27-s − 0.944·28-s + 1.29·29-s + 0.182·30-s + 0.179·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 886 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 886 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3211396228\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3211396228\) |
| \(L(\frac{3}{2})\) |
|
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$ | \( ( 1 - T )( 1 + p T + p T^{2} ) \) |
| 443 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 24 T + p T^{2} ) \) |
| good | 3 | $D_{4}$ | \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - T - p T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 5 T + 15 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 8 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 9 T + 43 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - T - 30 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T - 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 - T - 11 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + T - 17 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - T + 90 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $D_{4}$ | \( 1 + 2 T - 86 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 5 T + 90 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 12 T + 148 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 10 T + 40 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 9 T + 133 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
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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
−19.6344996221, −19.1213111747, −18.4607813800, −17.9186978829, −17.4607106027, −16.8141510035, −16.1601933266, −15.8418580941, −15.5545248221, −14.2966698665, −13.5299754272, −13.1001390515, −12.6426871380, −11.5532274637, −11.0432855275, −10.3516095653, −9.73381376453, −9.11765167357, −8.35741768242, −7.00282697910, −6.65662143905, −5.87534085956, −4.51401815793, −2.84240062922,
2.84240062922, 4.51401815793, 5.87534085956, 6.65662143905, 7.00282697910, 8.35741768242, 9.11765167357, 9.73381376453, 10.3516095653, 11.0432855275, 11.5532274637, 12.6426871380, 13.1001390515, 13.5299754272, 14.2966698665, 15.5545248221, 15.8418580941, 16.1601933266, 16.8141510035, 17.4607106027, 17.9186978829, 18.4607813800, 19.1213111747, 19.6344996221
