| L(s) = 1 | − 2·2-s − 2·3-s + 4·6-s − 2·7-s + 4·8-s − 3·9-s − 2·11-s − 12·13-s + 4·14-s − 4·16-s − 2·17-s + 6·18-s − 6·19-s + 4·21-s + 4·22-s + 12·23-s − 8·24-s + 25-s + 24·26-s + 14·27-s − 10·29-s + 4·33-s + 4·34-s − 6·37-s + 12·38-s + 24·39-s − 6·41-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 1.63·6-s − 0.755·7-s + 1.41·8-s − 9-s − 0.603·11-s − 3.32·13-s + 1.06·14-s − 16-s − 0.485·17-s + 1.41·18-s − 1.37·19-s + 0.872·21-s + 0.852·22-s + 2.50·23-s − 1.63·24-s + 1/5·25-s + 4.70·26-s + 2.69·27-s − 1.85·29-s + 0.696·33-s + 0.685·34-s − 0.986·37-s + 1.94·38-s + 3.84·39-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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_1$ | \( ( 1 - T )( 1 + T ) \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−16.6540047341, −15.6472314777, −15.0024513462, −14.5908663149, −14.5223487556, −13.2928378761, −13.2171078373, −12.5349234542, −12.2076238369, −11.5217209825, −11.0949806319, −10.4049849179, −10.3137986772, −9.45854299201, −9.28853187748, −8.68481047931, −8.23209814214, −7.31046477239, −7.10940075033, −6.36900778287, −5.44890211750, −4.92691899598, −4.74954893087, −3.14330660201, −2.37505881343, 0, 0,
2.37505881343, 3.14330660201, 4.74954893087, 4.92691899598, 5.44890211750, 6.36900778287, 7.10940075033, 7.31046477239, 8.23209814214, 8.68481047931, 9.28853187748, 9.45854299201, 10.3137986772, 10.4049849179, 11.0949806319, 11.5217209825, 12.2076238369, 12.5349234542, 13.2171078373, 13.2928378761, 14.5223487556, 14.5908663149, 15.0024513462, 15.6472314777, 16.6540047341
