| L(s) = 1 | − 3-s + 2·4-s − 5-s + 11-s − 2·12-s + 13-s + 15-s + 3·16-s + 17-s − 2·20-s + 27-s − 2·29-s − 33-s − 39-s + 2·44-s − 2·47-s − 3·48-s − 51-s + 2·52-s − 55-s + 2·60-s + 4·64-s − 65-s + 2·68-s − 2·71-s + 73-s − 2·79-s + ⋯ |
| L(s) = 1 | − 3-s + 2·4-s − 5-s + 11-s − 2·12-s + 13-s + 15-s + 3·16-s + 17-s − 2·20-s + 27-s − 2·29-s − 33-s − 39-s + 2·44-s − 2·47-s − 3·48-s − 51-s + 2·52-s − 55-s + 2·60-s + 4·64-s − 65-s + 2·68-s − 2·71-s + 73-s − 2·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.362499655\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.362499655\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
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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
−9.617165939713496771234483052794, −8.873277321730015430779078568338, −8.668346314552491140691315574926, −8.034568816534732898264124423411, −7.67328203531851811377422707575, −7.63618700609401585382443780578, −6.96699070314400794099360902047, −6.57620383145966869357765757476, −6.51989606614620395412128172753, −5.87904768139759362611052414295, −5.58948473666710481326945648670, −5.44115461187505276592197455638, −4.59234477865635825934491422907, −4.03672615170658931443114839217, −3.65357684221529865691539372548, −3.11461549321011374022210516117, −3.00316903791872139917304019375, −1.82916672696701128280875576354, −1.69606744992959541903140378407, −0.900154250872804777173279294693,
0.900154250872804777173279294693, 1.69606744992959541903140378407, 1.82916672696701128280875576354, 3.00316903791872139917304019375, 3.11461549321011374022210516117, 3.65357684221529865691539372548, 4.03672615170658931443114839217, 4.59234477865635825934491422907, 5.44115461187505276592197455638, 5.58948473666710481326945648670, 5.87904768139759362611052414295, 6.51989606614620395412128172753, 6.57620383145966869357765757476, 6.96699070314400794099360902047, 7.63618700609401585382443780578, 7.67328203531851811377422707575, 8.034568816534732898264124423411, 8.668346314552491140691315574926, 8.873277321730015430779078568338, 9.617165939713496771234483052794
