| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 4·7-s − 8-s + 9-s − 4·11-s + 12-s + 4·14-s + 16-s + 4·17-s − 18-s − 4·21-s + 4·22-s − 24-s + 27-s − 4·28-s − 32-s − 4·33-s − 4·34-s + 36-s + 4·42-s + 8·43-s − 4·44-s + 48-s − 2·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s − 1.20·11-s + 0.288·12-s + 1.06·14-s + 1/4·16-s + 0.970·17-s − 0.235·18-s − 0.872·21-s + 0.852·22-s − 0.204·24-s + 0.192·27-s − 0.755·28-s − 0.176·32-s − 0.696·33-s − 0.685·34-s + 1/6·36-s + 0.617·42-s + 1.21·43-s − 0.603·44-s + 0.144·48-s − 2/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 540000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_1$ | \( 1 - T \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−8.236259450125909594320955817325, −7.79436266718632723136861068704, −7.34830902118093822137916374938, −7.19091277442794332068165955627, −6.39955394184406631525545593399, −6.04674329942688851458805581356, −5.66765515312328094809085573246, −4.95720185791601620138850804643, −4.40421151945420287303938949967, −3.43332661175523197849238647019, −3.36984414779728341388392819787, −2.66426897593599555100997020524, −2.17693746121355235992569926706, −1.09380620717066232730427473522, 0,
1.09380620717066232730427473522, 2.17693746121355235992569926706, 2.66426897593599555100997020524, 3.36984414779728341388392819787, 3.43332661175523197849238647019, 4.40421151945420287303938949967, 4.95720185791601620138850804643, 5.66765515312328094809085573246, 6.04674329942688851458805581356, 6.39955394184406631525545593399, 7.19091277442794332068165955627, 7.34830902118093822137916374938, 7.79436266718632723136861068704, 8.236259450125909594320955817325
