| L(s) = 1 | − 3·2-s − 2·3-s + 4·4-s − 3·5-s + 6·6-s − 5·7-s − 3·8-s + 4·9-s + 9·10-s − 7·11-s − 8·12-s − 4·13-s + 15·14-s + 6·15-s + 3·16-s − 17-s − 12·18-s − 2·19-s − 12·20-s + 10·21-s + 21·22-s − 7·23-s + 6·24-s + 2·25-s + 12·26-s − 5·27-s − 20·28-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.15·3-s + 2·4-s − 1.34·5-s + 2.44·6-s − 1.88·7-s − 1.06·8-s + 4/3·9-s + 2.84·10-s − 2.11·11-s − 2.30·12-s − 1.10·13-s + 4.00·14-s + 1.54·15-s + 3/4·16-s − 0.242·17-s − 2.82·18-s − 0.458·19-s − 2.68·20-s + 2.18·21-s + 4.47·22-s − 1.45·23-s + 1.22·24-s + 2/5·25-s + 2.35·26-s − 0.962·27-s − 3.77·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10005 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10005 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T + p T^{2} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 3 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 9 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + T + 8 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - T - 8 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 3 T + 16 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T - 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $D_{4}$ | \( 1 + 5 T + 112 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $D_{4}$ | \( 1 - 5 T + 56 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 18 T + 166 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 2 T - 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 17 T + 188 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 6 T - 42 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−17.0305853597, −16.6220522871, −16.3134047927, −15.8323722713, −15.4888692302, −15.3008590699, −14.1896249638, −13.1750842647, −12.8383387071, −12.4791925880, −11.9438562267, −11.2814501675, −10.6948802146, −10.1685117182, −9.85229374616, −9.62788712882, −8.63308358868, −8.04745725326, −7.59062894660, −7.18547537930, −6.39263081412, −5.70151241334, −4.75083207653, −3.77938542657, −2.61433593162, 0, 0,
2.61433593162, 3.77938542657, 4.75083207653, 5.70151241334, 6.39263081412, 7.18547537930, 7.59062894660, 8.04745725326, 8.63308358868, 9.62788712882, 9.85229374616, 10.1685117182, 10.6948802146, 11.2814501675, 11.9438562267, 12.4791925880, 12.8383387071, 13.1750842647, 14.1896249638, 15.3008590699, 15.4888692302, 15.8323722713, 16.3134047927, 16.6220522871, 17.0305853597
