| L(s) = 1 | − 4·2-s + 8·4-s + 3·7-s − 8·8-s + 9-s + 4·11-s − 12·14-s − 4·16-s − 4·18-s − 16·22-s − 12·23-s + 24·28-s + 20·29-s + 32·32-s + 8·36-s − 4·37-s − 2·43-s + 32·44-s + 48·46-s + 2·49-s + 8·53-s − 24·56-s − 80·58-s + 3·63-s − 64·64-s + 6·67-s − 16·71-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 4·4-s + 1.13·7-s − 2.82·8-s + 1/3·9-s + 1.20·11-s − 3.20·14-s − 16-s − 0.942·18-s − 3.41·22-s − 2.50·23-s + 4.53·28-s + 3.71·29-s + 5.65·32-s + 4/3·36-s − 0.657·37-s − 0.304·43-s + 4.82·44-s + 7.07·46-s + 2/7·49-s + 1.09·53-s − 3.20·56-s − 10.5·58-s + 0.377·63-s − 8·64-s + 0.733·67-s − 1.89·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4950734526\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4950734526\) |
| \(L(\frac{3}{2})\) |
|
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_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
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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
−8.655074250265874594654667643843, −8.574576876976377679990287397231, −8.114757177440921419069207299350, −7.889167498214326068215591164801, −7.24727019658266874226795795729, −6.77287251257420576381216593643, −6.47848947541388825254209524969, −5.78543755363694734638896669447, −4.75926271769308777404655681076, −4.48018036005395779311676580836, −3.93239261316981647030035469500, −2.69932188173779771903247956783, −1.90319955654708104595670502735, −1.47286537697639516097126114886, −0.73191907264073361464195960844,
0.73191907264073361464195960844, 1.47286537697639516097126114886, 1.90319955654708104595670502735, 2.69932188173779771903247956783, 3.93239261316981647030035469500, 4.48018036005395779311676580836, 4.75926271769308777404655681076, 5.78543755363694734638896669447, 6.47848947541388825254209524969, 6.77287251257420576381216593643, 7.24727019658266874226795795729, 7.889167498214326068215591164801, 8.114757177440921419069207299350, 8.574576876976377679990287397231, 8.655074250265874594654667643843
