| L(s) = 1 | − 2·2-s − 3·3-s + 2·4-s − 2·5-s + 6·6-s − 7-s − 4·8-s + 2·9-s + 4·10-s − 6·12-s − 13-s + 2·14-s + 6·15-s + 8·16-s − 4·18-s + 19-s − 4·20-s + 3·21-s + 23-s + 12·24-s − 2·25-s + 2·26-s + 6·27-s − 2·28-s − 5·29-s − 12·30-s − 8·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.73·3-s + 4-s − 0.894·5-s + 2.44·6-s − 0.377·7-s − 1.41·8-s + 2/3·9-s + 1.26·10-s − 1.73·12-s − 0.277·13-s + 0.534·14-s + 1.54·15-s + 2·16-s − 0.942·18-s + 0.229·19-s − 0.894·20-s + 0.654·21-s + 0.208·23-s + 2.44·24-s − 2/5·25-s + 0.392·26-s + 1.15·27-s − 0.377·28-s − 0.928·29-s − 2.19·30-s − 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1385 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1385 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + T + p T^{2} ) \) |
| 277 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 24 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + T + 9 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + T - 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - T + 18 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 5 T + 53 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 9 T + 65 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 9 T + 47 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 71 | $D_{4}$ | \( 1 - 12 T + 106 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 11 T + 152 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 6 T + 78 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 8 T + 90 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $D_{4}$ | \( 1 - 10 T + 84 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
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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
−19.6273643581, −18.8639617108, −18.5671888805, −18.0220939371, −17.4836945764, −16.9995768034, −16.8168020807, −16.0736788406, −15.5842115729, −15.0360764322, −14.2963698968, −13.1987788447, −12.3245514773, −12.0211525907, −11.4317925881, −11.0731814029, −10.2666941419, −9.63078676148, −8.91530033341, −8.16968598156, −7.42758039961, −6.47110463922, −5.90223324063, −5.00367531507, −3.36744843913, 0,
3.36744843913, 5.00367531507, 5.90223324063, 6.47110463922, 7.42758039961, 8.16968598156, 8.91530033341, 9.63078676148, 10.2666941419, 11.0731814029, 11.4317925881, 12.0211525907, 12.3245514773, 13.1987788447, 14.2963698968, 15.0360764322, 15.5842115729, 16.0736788406, 16.8168020807, 16.9995768034, 17.4836945764, 18.0220939371, 18.5671888805, 18.8639617108, 19.6273643581
