| L(s) = 1 | + 2-s − 2·5-s + 4·7-s − 8-s − 2·10-s − 10·11-s + 4·14-s − 16-s − 4·17-s − 8·19-s − 10·22-s + 8·23-s − 7·25-s − 5·29-s − 3·31-s − 4·34-s − 8·35-s + 4·37-s − 8·38-s + 2·40-s − 2·43-s + 8·46-s − 6·47-s + 9·49-s − 7·50-s − 9·53-s + 20·55-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.894·5-s + 1.51·7-s − 0.353·8-s − 0.632·10-s − 3.01·11-s + 1.06·14-s − 1/4·16-s − 0.970·17-s − 1.83·19-s − 2.13·22-s + 1.66·23-s − 7/5·25-s − 0.928·29-s − 0.538·31-s − 0.685·34-s − 1.35·35-s + 0.657·37-s − 1.29·38-s + 0.316·40-s − 0.304·43-s + 1.17·46-s − 0.875·47-s + 9/7·49-s − 0.989·50-s − 1.23·53-s + 2.69·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5436923444\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5436923444\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 5 T - 4 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 2 T - 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 11 T + 62 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 3 T - 70 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 7 T - 34 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 7 T - 48 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
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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
−10.33561829894124781275833330163, −9.539724846620569600839619587997, −9.193709751118144522342267709656, −8.455097941681831741228575242327, −8.350707281662787806307649899356, −7.85306818704118945017810025630, −7.83576576311369556179647744827, −7.05555626716470911464647289123, −6.94158105914127859257004100464, −5.84497064628903122383256838000, −5.78255094467497370465565471221, −5.22168614633782176726790053995, −4.73545741747289084490952996941, −4.40424962104469368739024281006, −4.25235097662830384124164097681, −3.20587130788720384140379930745, −2.97184717759455502003556207532, −2.04739439191801017158504118707, −1.93831245664780331247692642475, −0.26708513409587534597085687755,
0.26708513409587534597085687755, 1.93831245664780331247692642475, 2.04739439191801017158504118707, 2.97184717759455502003556207532, 3.20587130788720384140379930745, 4.25235097662830384124164097681, 4.40424962104469368739024281006, 4.73545741747289084490952996941, 5.22168614633782176726790053995, 5.78255094467497370465565471221, 5.84497064628903122383256838000, 6.94158105914127859257004100464, 7.05555626716470911464647289123, 7.83576576311369556179647744827, 7.85306818704118945017810025630, 8.350707281662787806307649899356, 8.455097941681831741228575242327, 9.193709751118144522342267709656, 9.539724846620569600839619587997, 10.33561829894124781275833330163
