L(s) = 1 | − 2·2-s − 3·3-s + 4-s − 3·5-s + 6·6-s − 4·7-s + 4·9-s + 6·10-s − 3·12-s − 9·13-s + 8·14-s + 9·15-s + 16-s − 17-s − 8·18-s − 2·19-s − 3·20-s + 12·21-s − 6·23-s + 5·25-s + 18·26-s − 4·28-s + 4·29-s − 18·30-s − 2·31-s + 2·32-s + 2·34-s + ⋯ |
L(s) = 1 | − 1.41·2-s − 1.73·3-s + 1/2·4-s − 1.34·5-s + 2.44·6-s − 1.51·7-s + 4/3·9-s + 1.89·10-s − 0.866·12-s − 2.49·13-s + 2.13·14-s + 2.32·15-s + 1/4·16-s − 0.242·17-s − 1.88·18-s − 0.458·19-s − 0.670·20-s + 2.61·21-s − 1.25·23-s + 25-s + 3.53·26-s − 0.755·28-s + 0.742·29-s − 3.28·30-s − 0.359·31-s + 0.353·32-s + 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9771 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9771 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| 3257 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 42 T + p T^{2} ) \) |
good | 2 | $D_{4}$ | \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 49 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T - p T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 10 T + 82 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + T + 60 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 72 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 10 T + 97 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - T + 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 16 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 18 T + 192 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - T + 92 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 18 T + 187 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 2 T + 62 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 2 T + 50 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
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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
−17.2590093325, −16.7763123955, −16.2169561575, −16.0198635677, −15.5086970488, −14.8556945919, −14.3687723610, −13.4138077971, −12.7162123620, −12.2019329232, −12.0347879712, −11.6978179686, −10.8132485252, −10.3009718480, −9.87778390942, −9.58727224607, −8.63233969503, −8.27395653010, −7.25877091472, −7.04961990411, −6.37160296631, −5.54192015431, −4.80395821956, −4.02156733105, −2.80946997545, 0, 0,
2.80946997545, 4.02156733105, 4.80395821956, 5.54192015431, 6.37160296631, 7.04961990411, 7.25877091472, 8.27395653010, 8.63233969503, 9.58727224607, 9.87778390942, 10.3009718480, 10.8132485252, 11.6978179686, 12.0347879712, 12.2019329232, 12.7162123620, 13.4138077971, 14.3687723610, 14.8556945919, 15.5086970488, 16.0198635677, 16.2169561575, 16.7763123955, 17.2590093325