| L(s) = 1 | − 2-s − 3-s − 2·5-s + 6-s + 8-s + 2·10-s − 2·13-s + 2·15-s − 16-s + 19-s − 2·23-s − 24-s + 25-s + 2·26-s + 27-s − 2·30-s − 38-s + 2·39-s − 2·40-s + 2·46-s + 48-s − 50-s − 54-s − 57-s − 2·59-s + 61-s + 64-s + ⋯ |
| L(s) = 1 | − 2-s − 3-s − 2·5-s + 6-s + 8-s + 2·10-s − 2·13-s + 2·15-s − 16-s + 19-s − 2·23-s − 24-s + 25-s + 2·26-s + 27-s − 2·30-s − 38-s + 2·39-s − 2·40-s + 2·46-s + 48-s − 50-s − 54-s − 57-s − 2·59-s + 61-s + 64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.0008032382988\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0008032382988\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−8.678823072014999364670049501212, −8.605852893270088321251759999521, −8.018444713588179514722949124677, −7.88107479113781708540118643740, −7.56205853281525838035978625803, −7.19759269325710739546937599720, −7.05192716179111697690513014297, −6.44611536824688367775281278041, −5.77551784126851747780892134716, −5.71654111896407470516932760850, −5.01650576265853764239350981387, −4.74157142594756227394563381850, −4.31062343527203027913334724230, −4.12842748159661787608799036502, −3.50890274371951980866263204186, −3.04963581831301415464755874808, −2.42539684432091705781786939488, −1.83147568710794522010079156713, −1.03978766618099246977404762834, −0.02515613220077678073874183418,
0.02515613220077678073874183418, 1.03978766618099246977404762834, 1.83147568710794522010079156713, 2.42539684432091705781786939488, 3.04963581831301415464755874808, 3.50890274371951980866263204186, 4.12842748159661787608799036502, 4.31062343527203027913334724230, 4.74157142594756227394563381850, 5.01650576265853764239350981387, 5.71654111896407470516932760850, 5.77551784126851747780892134716, 6.44611536824688367775281278041, 7.05192716179111697690513014297, 7.19759269325710739546937599720, 7.56205853281525838035978625803, 7.88107479113781708540118643740, 8.018444713588179514722949124677, 8.605852893270088321251759999521, 8.678823072014999364670049501212
