| L(s) = 1 | − 2-s − 3·3-s − 2·4-s − 2·5-s + 3·6-s − 2·7-s + 3·8-s + 4·9-s + 2·10-s + 4·11-s + 6·12-s − 4·13-s + 2·14-s + 6·15-s + 16-s − 17-s − 4·18-s − 7·19-s + 4·20-s + 6·21-s − 4·22-s + 3·23-s − 9·24-s − 25-s + 4·26-s + 4·28-s − 2·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s − 4-s − 0.894·5-s + 1.22·6-s − 0.755·7-s + 1.06·8-s + 4/3·9-s + 0.632·10-s + 1.20·11-s + 1.73·12-s − 1.10·13-s + 0.534·14-s + 1.54·15-s + 1/4·16-s − 0.242·17-s − 0.942·18-s − 1.60·19-s + 0.894·20-s + 1.30·21-s − 0.852·22-s + 0.625·23-s − 1.83·24-s − 1/5·25-s + 0.784·26-s + 0.755·28-s − 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1497 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1497 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| 499 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 8 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 2 T + p T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + p T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + T - 13 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 19 | $C_4$ | \( 1 + 7 T + 46 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 3 T - 3 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $D_{4}$ | \( 1 - 2 T - 28 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 8 T + p T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 9 T + 45 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + T - 35 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2 T + 28 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2 T + 75 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 7 T + 43 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 12 T + 112 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 13 T + 127 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - T + 77 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 14 T + 150 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 53 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 10 T + 198 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−19.2855817528, −19.0615980890, −18.5445714241, −17.6180248610, −17.4562614834, −16.9681486433, −16.7375346436, −15.9792877794, −15.3172049980, −14.6806868407, −13.9905507779, −13.0370680405, −12.6795417955, −11.9037299845, −11.6360820210, −10.8274180676, −10.1278529634, −9.62960078848, −8.76919069981, −8.28476716690, −6.92793088928, −6.68987340027, −5.49965237248, −4.65509555918, −3.89805511854, 0,
3.89805511854, 4.65509555918, 5.49965237248, 6.68987340027, 6.92793088928, 8.28476716690, 8.76919069981, 9.62960078848, 10.1278529634, 10.8274180676, 11.6360820210, 11.9037299845, 12.6795417955, 13.0370680405, 13.9905507779, 14.6806868407, 15.3172049980, 15.9792877794, 16.7375346436, 16.9681486433, 17.4562614834, 17.6180248610, 18.5445714241, 19.0615980890, 19.2855817528
