| L(s) = 1 | + 2·2-s − 2·3-s + 3·4-s − 2·5-s − 4·6-s + 2·7-s + 4·8-s + 3·9-s − 4·10-s + 2·11-s − 6·12-s − 2·13-s + 4·14-s + 4·15-s + 5·16-s − 2·17-s + 6·18-s − 2·19-s − 6·20-s − 4·21-s + 4·22-s − 6·23-s − 8·24-s + 3·25-s − 4·26-s − 4·27-s + 6·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 3/2·4-s − 0.894·5-s − 1.63·6-s + 0.755·7-s + 1.41·8-s + 9-s − 1.26·10-s + 0.603·11-s − 1.73·12-s − 0.554·13-s + 1.06·14-s + 1.03·15-s + 5/4·16-s − 0.485·17-s + 1.41·18-s − 0.458·19-s − 1.34·20-s − 0.872·21-s + 0.852·22-s − 1.25·23-s − 1.63·24-s + 3/5·25-s − 0.784·26-s − 0.769·27-s + 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31472100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31472100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| 17 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_4$ | \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_4$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 82 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 2 T + 102 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 2 T + 114 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 146 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 2 T + 142 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 6 T + 122 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 4 T + 102 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
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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
−7.72720140769571260090271352994, −7.44726748742563166770616806118, −7.06320305733496384992048359302, −6.92008364183732011081629801422, −6.30743467368498739481516435414, −6.30219638517839186003059801060, −5.50646064308060638903556212029, −5.47419669463175265273752797999, −4.94472479330928166072570456445, −4.90684205222900227494035500445, −4.12702060115289826747020434403, −4.06183288310022029784484356589, −3.69348550174171394430188070865, −3.39695480961263546610750639941, −2.55099007706268509600220415376, −2.15300244560644359040990580256, −1.60966273223528092606398841137, −1.34465594250864819475464396983, 0, 0,
1.34465594250864819475464396983, 1.60966273223528092606398841137, 2.15300244560644359040990580256, 2.55099007706268509600220415376, 3.39695480961263546610750639941, 3.69348550174171394430188070865, 4.06183288310022029784484356589, 4.12702060115289826747020434403, 4.90684205222900227494035500445, 4.94472479330928166072570456445, 5.47419669463175265273752797999, 5.50646064308060638903556212029, 6.30219638517839186003059801060, 6.30743467368498739481516435414, 6.92008364183732011081629801422, 7.06320305733496384992048359302, 7.44726748742563166770616806118, 7.72720140769571260090271352994
