| L(s) = 1 | − 3·2-s − 4·3-s + 4·4-s − 5·5-s + 12·6-s − 7·7-s − 3·8-s + 8·9-s + 15·10-s − 8·11-s − 16·12-s − 7·13-s + 21·14-s + 20·15-s + 3·16-s − 6·17-s − 24·18-s − 6·19-s − 20·20-s + 28·21-s + 24·22-s − 8·23-s + 12·24-s + 12·25-s + 21·26-s − 12·27-s − 28·28-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 2.30·3-s + 2·4-s − 2.23·5-s + 4.89·6-s − 2.64·7-s − 1.06·8-s + 8/3·9-s + 4.74·10-s − 2.41·11-s − 4.61·12-s − 1.94·13-s + 5.61·14-s + 5.16·15-s + 3/4·16-s − 1.45·17-s − 5.65·18-s − 1.37·19-s − 4.47·20-s + 6.11·21-s + 5.11·22-s − 1.66·23-s + 2.44·24-s + 12/5·25-s + 4.11·26-s − 2.30·27-s − 5.29·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 440509 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 440509 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 440509 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 372 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + p T + 13 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $D_{4}$ | \( 1 + 7 T + 33 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 49 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 3 T + 10 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 67 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 3 T + 49 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 7 T + 71 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 2 T + 96 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 13 T + 91 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 2 T + 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2 T + 81 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 7 T + 103 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 68 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 6 T + 114 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6 T + 21 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 9 T + 76 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
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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
−13.1546268205, −12.7880525857, −12.5346839319, −12.1766719253, −11.8613110492, −11.4735081605, −10.9086661966, −10.5620030975, −10.2546157808, −10.0986322329, −9.64174940648, −9.08334175994, −8.59820322965, −7.96663039809, −7.77768501228, −7.31281302987, −6.88622844099, −6.49087534970, −6.01326952024, −5.46419066285, −4.82036564507, −4.37683859034, −3.67684006544, −2.94634803659, −2.25091762850, 0, 0, 0, 0,
2.25091762850, 2.94634803659, 3.67684006544, 4.37683859034, 4.82036564507, 5.46419066285, 6.01326952024, 6.49087534970, 6.88622844099, 7.31281302987, 7.77768501228, 7.96663039809, 8.59820322965, 9.08334175994, 9.64174940648, 10.0986322329, 10.2546157808, 10.5620030975, 10.9086661966, 11.4735081605, 11.8613110492, 12.1766719253, 12.5346839319, 12.7880525857, 13.1546268205
