L(s) = 1 | + 2·3-s + 2·4-s + 2·7-s + 3·9-s + 4·12-s + 4·13-s − 6·17-s + 4·19-s + 4·21-s + 12·23-s − 4·25-s + 4·27-s + 4·28-s − 6·29-s + 16·31-s + 6·36-s − 14·37-s + 8·39-s + 6·41-s + 4·43-s − 24·47-s + 3·49-s − 12·51-s + 8·52-s − 6·53-s + 8·57-s + 4·61-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 4-s + 0.755·7-s + 9-s + 1.15·12-s + 1.10·13-s − 1.45·17-s + 0.917·19-s + 0.872·21-s + 2.50·23-s − 4/5·25-s + 0.769·27-s + 0.755·28-s − 1.11·29-s + 2.87·31-s + 36-s − 2.30·37-s + 1.28·39-s + 0.937·41-s + 0.609·43-s − 3.50·47-s + 3/7·49-s − 1.68·51-s + 1.10·52-s − 0.824·53-s + 1.05·57-s + 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7112889 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7112889 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(7.026467746\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.026467746\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 127 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 37 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 36 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 43 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 85 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 91 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 112 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 120 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T - 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 12 T + 172 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 108 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2 T + 135 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 10 T + 165 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
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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
−9.002106697574846549238966804383, −8.582961843504661814972934814529, −8.333854013153734751896655491834, −7.82163518079327196831331359507, −7.74851595444853039329574339737, −7.00771028189804565477611062757, −6.88236497167738781734271983423, −6.43796658552937101059481167412, −6.29492396907212322616902875580, −5.45383735625270982849180921939, −4.98718079212420576325213364252, −4.82794472560227298101268428262, −4.19326088927695085362579266290, −3.76272811892513590923826996963, −3.13258262158209135687308174415, −2.97568520213590850281150463075, −2.43696633240949063460412632614, −1.68441610522505223825387644787, −1.66170372175561095069064860899, −0.796797708794726098704515066891,
0.796797708794726098704515066891, 1.66170372175561095069064860899, 1.68441610522505223825387644787, 2.43696633240949063460412632614, 2.97568520213590850281150463075, 3.13258262158209135687308174415, 3.76272811892513590923826996963, 4.19326088927695085362579266290, 4.82794472560227298101268428262, 4.98718079212420576325213364252, 5.45383735625270982849180921939, 6.29492396907212322616902875580, 6.43796658552937101059481167412, 6.88236497167738781734271983423, 7.00771028189804565477611062757, 7.74851595444853039329574339737, 7.82163518079327196831331359507, 8.333854013153734751896655491834, 8.582961843504661814972934814529, 9.002106697574846549238966804383