| L(s) = 1 | − 2-s + 2·4-s − 4·5-s + 2·7-s − 5·8-s + 4·10-s − 11-s + 2·13-s − 2·14-s + 5·16-s − 4·17-s − 12·19-s − 8·20-s + 22-s + 4·23-s + 5·25-s − 2·26-s + 4·28-s − 6·29-s − 4·31-s − 10·32-s + 4·34-s − 8·35-s − 12·37-s + 12·38-s + 20·40-s − 10·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 4-s − 1.78·5-s + 0.755·7-s − 1.76·8-s + 1.26·10-s − 0.301·11-s + 0.554·13-s − 0.534·14-s + 5/4·16-s − 0.970·17-s − 2.75·19-s − 1.78·20-s + 0.213·22-s + 0.834·23-s + 25-s − 0.392·26-s + 0.755·28-s − 1.11·29-s − 0.718·31-s − 1.76·32-s + 0.685·34-s − 1.35·35-s − 1.97·37-s + 1.94·38-s + 3.16·40-s − 1.56·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 793881 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 793881 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 11 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 10 T + 59 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 6 T - 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−10.06008018455552287803754459286, −9.207948177968772448133856251722, −8.832611423657132076547209976130, −8.726524999166781399492757690371, −8.198791041195620481261305819974, −8.017545465213814264107249330539, −7.43614320171919205216045652117, −6.90755415708715459241708014316, −6.54760725323309604010972060240, −6.42754529641153022576734659541, −5.33660393000965542334658859514, −5.26459297329502689742601361636, −4.36332847499736639683711378308, −3.87005922360318111365880062926, −3.59653643658300951519695607133, −2.88093968080214712478354047342, −2.08796912368310969291413397023, −1.70521221475953509159235261326, 0, 0,
1.70521221475953509159235261326, 2.08796912368310969291413397023, 2.88093968080214712478354047342, 3.59653643658300951519695607133, 3.87005922360318111365880062926, 4.36332847499736639683711378308, 5.26459297329502689742601361636, 5.33660393000965542334658859514, 6.42754529641153022576734659541, 6.54760725323309604010972060240, 6.90755415708715459241708014316, 7.43614320171919205216045652117, 8.017545465213814264107249330539, 8.198791041195620481261305819974, 8.726524999166781399492757690371, 8.832611423657132076547209976130, 9.207948177968772448133856251722, 10.06008018455552287803754459286
