| L(s) = 1 | − 2-s − 3-s − 4-s − 2·5-s + 6-s − 4·7-s + 8-s + 2·10-s + 12-s − 4·13-s + 4·14-s + 2·15-s + 3·16-s + 2·17-s − 3·19-s + 2·20-s + 4·21-s − 24-s − 2·25-s + 4·26-s + 4·27-s + 4·28-s + 29-s − 2·30-s − 31-s − 3·32-s − 2·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.894·5-s + 0.408·6-s − 1.51·7-s + 0.353·8-s + 0.632·10-s + 0.288·12-s − 1.10·13-s + 1.06·14-s + 0.516·15-s + 3/4·16-s + 0.485·17-s − 0.688·19-s + 0.447·20-s + 0.872·21-s − 0.204·24-s − 2/5·25-s + 0.784·26-s + 0.769·27-s + 0.755·28-s + 0.185·29-s − 0.365·30-s − 0.179·31-s − 0.530·32-s − 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3462 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3462 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| 577 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 2 T + p T^{2} ) \) |
| good | 5 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_4$ | \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 16 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 3 T + 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 40 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - T - 36 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 16 T + 140 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + T + 70 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6 T + 48 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 6 T + 52 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 12 T + 78 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 9 T + 18 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 5 T + 60 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 174 T^{2} + p^{2} T^{4} \) |
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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
−18.3125898413, −17.7552794033, −17.2284534301, −16.7721742765, −16.3120862578, −15.9784949389, −15.0766280128, −14.8760593503, −14.1015436320, −13.2626363213, −12.8376362780, −12.3067734969, −11.8408385091, −11.2195260964, −10.1716693840, −10.0449799715, −9.46046439174, −8.63620513209, −8.13035610494, −7.30588472939, −6.66118907662, −5.86240554515, −4.94984354053, −3.96773947126, −3.03407658337, 0,
3.03407658337, 3.96773947126, 4.94984354053, 5.86240554515, 6.66118907662, 7.30588472939, 8.13035610494, 8.63620513209, 9.46046439174, 10.0449799715, 10.1716693840, 11.2195260964, 11.8408385091, 12.3067734969, 12.8376362780, 13.2626363213, 14.1015436320, 14.8760593503, 15.0766280128, 15.9784949389, 16.3120862578, 16.7721742765, 17.2284534301, 17.7552794033, 18.3125898413
