| L(s) = 1 | + 4-s − 7-s + 9-s − 4·11-s + 12·13-s + 16-s + 4·17-s − 8·19-s + 16·23-s − 6·25-s − 28-s + 36-s − 20·37-s − 12·41-s − 4·44-s + 49-s + 12·52-s + 12·53-s + 12·61-s − 63-s + 64-s + 8·67-s + 4·68-s + 16·71-s + 20·73-s − 8·76-s + 4·77-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 0.377·7-s + 1/3·9-s − 1.20·11-s + 3.32·13-s + 1/4·16-s + 0.970·17-s − 1.83·19-s + 3.33·23-s − 6/5·25-s − 0.188·28-s + 1/6·36-s − 3.28·37-s − 1.87·41-s − 0.603·44-s + 1/7·49-s + 1.66·52-s + 1.64·53-s + 1.53·61-s − 0.125·63-s + 1/8·64-s + 0.977·67-s + 0.485·68-s + 1.89·71-s + 2.34·73-s − 0.917·76-s + 0.455·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1494108 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1494108 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.752999213\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.752999213\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$ | \( 1 + T \) |
| 11 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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
−8.209722972899758923826863695247, −7.36329223436070087794310634517, −6.84552480602986516155667793601, −6.61396756737736082376174832588, −6.46232054545852935313207935977, −5.53755315525818574410350601733, −5.33985014787602837985094112170, −5.14287919199214033627107465191, −4.05522826801866843177122831386, −3.62482887081886485478101246558, −3.49945591428017159980646979643, −2.83312116915105400792054837352, −2.03686460629757130002584503967, −1.48110886543319454700611992720, −0.75184239042057648945425095713,
0.75184239042057648945425095713, 1.48110886543319454700611992720, 2.03686460629757130002584503967, 2.83312116915105400792054837352, 3.49945591428017159980646979643, 3.62482887081886485478101246558, 4.05522826801866843177122831386, 5.14287919199214033627107465191, 5.33985014787602837985094112170, 5.53755315525818574410350601733, 6.46232054545852935313207935977, 6.61396756737736082376174832588, 6.84552480602986516155667793601, 7.36329223436070087794310634517, 8.209722972899758923826863695247
