| L(s) = 1 | − 3-s + 4-s − 4·5-s − 7-s + 9-s − 12-s + 4·15-s + 16-s + 4·17-s − 4·20-s + 21-s + 2·25-s − 27-s − 28-s + 4·35-s + 36-s − 20·37-s − 12·41-s − 8·43-s − 4·45-s − 48-s + 49-s − 4·51-s + 8·59-s + 4·60-s − 63-s + 64-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/2·4-s − 1.78·5-s − 0.377·7-s + 1/3·9-s − 0.288·12-s + 1.03·15-s + 1/4·16-s + 0.970·17-s − 0.894·20-s + 0.218·21-s + 2/5·25-s − 0.192·27-s − 0.188·28-s + 0.676·35-s + 1/6·36-s − 3.28·37-s − 1.87·41-s − 1.21·43-s − 0.596·45-s − 0.144·48-s + 1/7·49-s − 0.560·51-s + 1.04·59-s + 0.516·60-s − 0.125·63-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37044 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37044 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( 1 + T \) |
| 7 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−10.24612146790310983021844103291, −9.804201361737724343194425540841, −8.882543700432626086795887126363, −8.356918966632091922574869118017, −7.954464142207805330736993023418, −7.36399840290663473896741338506, −6.84552480602986516155667793601, −6.54855688605176886120203881032, −5.33985014787602837985094112170, −5.32455191890654195761811472253, −4.16943092436514571048096348777, −3.62482887081886485478101246558, −3.17941538941685381741709304651, −1.71238689353652455033144003099, 0,
1.71238689353652455033144003099, 3.17941538941685381741709304651, 3.62482887081886485478101246558, 4.16943092436514571048096348777, 5.32455191890654195761811472253, 5.33985014787602837985094112170, 6.54855688605176886120203881032, 6.84552480602986516155667793601, 7.36399840290663473896741338506, 7.954464142207805330736993023418, 8.356918966632091922574869118017, 8.882543700432626086795887126363, 9.804201361737724343194425540841, 10.24612146790310983021844103291
