| L(s) = 1 | − 2·2-s − 2·3-s + 2·4-s + 4·6-s − 4·7-s + 2·9-s + 2·11-s − 4·12-s − 2·13-s + 8·14-s − 4·16-s − 4·18-s − 6·19-s + 8·21-s − 4·22-s − 12·23-s + 4·26-s − 6·27-s − 8·28-s − 6·29-s − 16·31-s + 8·32-s − 4·33-s + 4·36-s + 6·37-s + 12·38-s + 4·39-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 4-s + 1.63·6-s − 1.51·7-s + 2/3·9-s + 0.603·11-s − 1.15·12-s − 0.554·13-s + 2.13·14-s − 16-s − 0.942·18-s − 1.37·19-s + 1.74·21-s − 0.852·22-s − 2.50·23-s + 0.784·26-s − 1.15·27-s − 1.51·28-s − 1.11·29-s − 2.87·31-s + 1.41·32-s − 0.696·33-s + 2/3·36-s + 0.986·37-s + 1.94·38-s + 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−10.85484159657115343683812389551, −10.62715082683469826464073059420, −9.868540195777612973233646985484, −9.826661866431389346779887392965, −9.365213889626572222592258006465, −9.000279385484718572602242679521, −8.239565037374621293852722793657, −7.895921455341888060815499456993, −7.16852640369284220072545235432, −6.97054748817549145268800107900, −6.24154080623458419056170767990, −6.04921895578873462321944675501, −5.50419091175507436436231964851, −4.66285355713816128156926393374, −3.86385342093561225876920179955, −3.63713117883414593857642957260, −2.21855833427292704335805704474, −1.77798499173127444852298188332, 0, 0,
1.77798499173127444852298188332, 2.21855833427292704335805704474, 3.63713117883414593857642957260, 3.86385342093561225876920179955, 4.66285355713816128156926393374, 5.50419091175507436436231964851, 6.04921895578873462321944675501, 6.24154080623458419056170767990, 6.97054748817549145268800107900, 7.16852640369284220072545235432, 7.895921455341888060815499456993, 8.239565037374621293852722793657, 9.000279385484718572602242679521, 9.365213889626572222592258006465, 9.826661866431389346779887392965, 9.868540195777612973233646985484, 10.62715082683469826464073059420, 10.85484159657115343683812389551
