| L(s) = 1 | − 2-s − 2·3-s − 2·4-s − 2·5-s + 2·6-s + 4·7-s + 3·8-s + 2·9-s + 2·10-s − 4·11-s + 4·12-s − 4·14-s + 4·15-s + 16-s + 6·17-s − 2·18-s + 4·20-s − 8·21-s + 4·22-s − 2·23-s − 6·24-s − 7·25-s − 6·27-s − 8·28-s + 10·29-s − 4·30-s − 2·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 4-s − 0.894·5-s + 0.816·6-s + 1.51·7-s + 1.06·8-s + 2/3·9-s + 0.632·10-s − 1.20·11-s + 1.15·12-s − 1.06·14-s + 1.03·15-s + 1/4·16-s + 1.45·17-s − 0.471·18-s + 0.894·20-s − 1.74·21-s + 0.852·22-s − 0.417·23-s − 1.22·24-s − 7/5·25-s − 1.15·27-s − 1.51·28-s + 1.85·29-s − 0.730·30-s − 0.359·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27447121 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27447121 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 13 | | \( 1 \) |
| 31 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $D_{4}$ | \( 1 - 4 T + 13 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 33 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 10 T + 78 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 82 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 12 T + 122 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 113 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 6 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 142 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 10 T + 138 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 122 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 10 T + 158 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 14 T + 163 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
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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
−7.83211803160838728625112178041, −7.73006025036561664417166231350, −7.69524998786710847089693807829, −7.21673154854124076930163973225, −6.54416460607427313252056096235, −6.11017368367730709074934194003, −5.71469709969348539926511345084, −5.53304576761223347255287967788, −4.92874215846122714861192680614, −4.78441934019968973245178148667, −4.51678375986888254829311760129, −4.16688118535922234686422391632, −3.41196578023567722059076075111, −3.34012128743553683590190266205, −2.53300331217918229502030644640, −1.76356488129589863312752519105, −1.48757630322314548107512981345, −0.841073734749130918055299836761, 0, 0,
0.841073734749130918055299836761, 1.48757630322314548107512981345, 1.76356488129589863312752519105, 2.53300331217918229502030644640, 3.34012128743553683590190266205, 3.41196578023567722059076075111, 4.16688118535922234686422391632, 4.51678375986888254829311760129, 4.78441934019968973245178148667, 4.92874215846122714861192680614, 5.53304576761223347255287967788, 5.71469709969348539926511345084, 6.11017368367730709074934194003, 6.54416460607427313252056096235, 7.21673154854124076930163973225, 7.69524998786710847089693807829, 7.73006025036561664417166231350, 7.83211803160838728625112178041
