| L(s) = 1 | − 2·2-s + 4-s − 4·5-s + 2·7-s + 8·10-s − 2·11-s + 4·13-s − 4·14-s + 16-s + 6·17-s − 6·19-s − 4·20-s + 4·22-s − 14·23-s + 2·25-s − 8·26-s + 2·28-s − 2·29-s + 2·31-s + 2·32-s − 12·34-s − 8·35-s − 4·37-s + 12·38-s − 4·41-s − 8·43-s − 2·44-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s − 1.78·5-s + 0.755·7-s + 2.52·10-s − 0.603·11-s + 1.10·13-s − 1.06·14-s + 1/4·16-s + 1.45·17-s − 1.37·19-s − 0.894·20-s + 0.852·22-s − 2.91·23-s + 2/5·25-s − 1.56·26-s + 0.377·28-s − 0.371·29-s + 0.359·31-s + 0.353·32-s − 2.05·34-s − 1.35·35-s − 0.657·37-s + 1.94·38-s − 0.624·41-s − 1.21·43-s − 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700569 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700569 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 31 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $D_{4}$ | \( 1 - 2 T + p T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 6 T + 39 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 2 T + 51 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 94 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 66 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2 T + 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 150 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 16 T + 178 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 18 T + 215 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 10 T + 131 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 20 T + 262 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
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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
−9.958279184910458237859401317733, −9.819752284087942399940637855564, −8.810208183561055438216786863558, −8.622692588848825480834782076823, −8.188904080829975944315058254174, −8.144742133736336763482408382306, −7.63270765001883914040026672045, −7.51617518721846602351759018782, −6.61405244104112092662514728392, −6.10249779804671346222671040950, −5.77708126495150555397736882429, −5.03958790914630250412141761589, −4.40415725460730876380170691043, −4.00940428856782464888216113183, −3.60796443423768488767789525730, −3.02288335287865288727240611315, −1.92039823879916862601999032853, −1.41929196600223562441970308150, 0, 0,
1.41929196600223562441970308150, 1.92039823879916862601999032853, 3.02288335287865288727240611315, 3.60796443423768488767789525730, 4.00940428856782464888216113183, 4.40415725460730876380170691043, 5.03958790914630250412141761589, 5.77708126495150555397736882429, 6.10249779804671346222671040950, 6.61405244104112092662514728392, 7.51617518721846602351759018782, 7.63270765001883914040026672045, 8.144742133736336763482408382306, 8.188904080829975944315058254174, 8.622692588848825480834782076823, 8.810208183561055438216786863558, 9.819752284087942399940637855564, 9.958279184910458237859401317733
