| L(s) = 1 | − 4·2-s + 12·4-s − 10·5-s − 32·8-s − 8·9-s + 40·10-s + 26·13-s + 80·16-s + 32·18-s − 120·20-s + 75·25-s − 104·26-s + 64·29-s − 192·32-s − 96·36-s + 112·37-s + 320·40-s + 80·45-s + 98·49-s − 300·50-s + 312·52-s − 256·58-s − 224·61-s + 448·64-s − 260·65-s + 256·72-s − 32·73-s + ⋯ |
| L(s) = 1 | − 2·2-s + 3·4-s − 2·5-s − 4·8-s − 8/9·9-s + 4·10-s + 2·13-s + 5·16-s + 16/9·18-s − 6·20-s + 3·25-s − 4·26-s + 2.20·29-s − 6·32-s − 8/3·36-s + 3.02·37-s + 8·40-s + 16/9·45-s + 2·49-s − 6·50-s + 6·52-s − 4.41·58-s − 3.67·61-s + 7·64-s − 4·65-s + 32/9·72-s − 0.438·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67600 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5203168778\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5203168778\) |
| \(L(2)\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 8 T^{2} + p^{4} T^{4} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 8 T^{2} + p^{4} T^{4} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 488 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 1048 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 32 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 1688 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 56 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 1592 T^{2} + p^{4} T^{4} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 6728 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 112 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 8872 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 16 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p^{2} T^{2} )^{2} \) |
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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
−11.57748614592028647083450366458, −11.51762556944495684334854312589, −10.86258968689969116049164702642, −10.75702099446751601803819861464, −10.26307773179102925046553016034, −9.295813325011582519159653296583, −9.040950820187250858068428515373, −8.492835141642424274516047782176, −8.165238478752738249047123430102, −7.944602909561799684455953456328, −7.31682359057672537640123533227, −6.78749766365069253829568875804, −6.02896472124368698735425157091, −5.98061548324157323558893162555, −4.58178722624985682925676851284, −3.89867167792356412653807616017, −3.01708134682481243206613288266, −2.80851709972388035157983360358, −1.19194363187592749218041321955, −0.60961221647217996458756053313,
0.60961221647217996458756053313, 1.19194363187592749218041321955, 2.80851709972388035157983360358, 3.01708134682481243206613288266, 3.89867167792356412653807616017, 4.58178722624985682925676851284, 5.98061548324157323558893162555, 6.02896472124368698735425157091, 6.78749766365069253829568875804, 7.31682359057672537640123533227, 7.944602909561799684455953456328, 8.165238478752738249047123430102, 8.492835141642424274516047782176, 9.040950820187250858068428515373, 9.295813325011582519159653296583, 10.26307773179102925046553016034, 10.75702099446751601803819861464, 10.86258968689969116049164702642, 11.51762556944495684334854312589, 11.57748614592028647083450366458
