| L(s) = 1 | − 2·2-s − 2·3-s − 3·5-s + 4·6-s − 7-s + 4·8-s + 9-s + 6·10-s − 2·11-s − 2·13-s + 2·14-s + 6·15-s − 4·16-s − 2·18-s + 2·19-s + 2·21-s + 4·22-s + 3·23-s − 8·24-s + 2·25-s + 4·26-s + 4·27-s − 5·29-s − 12·30-s − 12·31-s + 4·33-s + 3·35-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s − 1.34·5-s + 1.63·6-s − 0.377·7-s + 1.41·8-s + 1/3·9-s + 1.89·10-s − 0.603·11-s − 0.554·13-s + 0.534·14-s + 1.54·15-s − 16-s − 0.471·18-s + 0.458·19-s + 0.436·21-s + 0.852·22-s + 0.625·23-s − 1.63·24-s + 2/5·25-s + 0.784·26-s + 0.769·27-s − 0.928·29-s − 2.19·30-s − 2.15·31-s + 0.696·33-s + 0.507·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1269 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1269 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 9 T + p T^{2} ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $D_{4}$ | \( 1 + 3 T + 7 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + T - p T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 - 3 T + 19 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 5 T + 19 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 + 3 T - 17 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 9 T + 67 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 10 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 5 T + 48 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 2 T + 46 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 3 T + 40 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 5 T + 26 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2 T - 44 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - T - 26 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 3 T - 104 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
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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
−19.5776908390, −19.1327910738, −18.5651297850, −18.3417366688, −17.7297115042, −17.1254621518, −16.6829307484, −16.3905149442, −15.5777161957, −15.1541138027, −14.2850893251, −13.4099667552, −12.8006260468, −12.0863289288, −11.6019535248, −10.8177976648, −10.4563514276, −9.58611632056, −8.98024712989, −8.28216613941, −7.49310733454, −7.05207241730, −5.60448676445, −4.88015795500, −3.68093204633, 0,
3.68093204633, 4.88015795500, 5.60448676445, 7.05207241730, 7.49310733454, 8.28216613941, 8.98024712989, 9.58611632056, 10.4563514276, 10.8177976648, 11.6019535248, 12.0863289288, 12.8006260468, 13.4099667552, 14.2850893251, 15.1541138027, 15.5777161957, 16.3905149442, 16.6829307484, 17.1254621518, 17.7297115042, 18.3417366688, 18.5651297850, 19.1327910738, 19.5776908390
