| L(s) = 1 | − 3-s + 4-s + 9-s − 12-s + 4·13-s + 16-s − 2·19-s − 6·25-s − 27-s + 8·31-s + 36-s + 20·37-s − 4·39-s + 8·43-s − 48-s − 14·49-s + 4·52-s + 2·57-s + 28·61-s + 64-s − 24·67-s − 12·73-s + 6·75-s − 2·76-s − 8·79-s + 81-s − 8·93-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/2·4-s + 1/3·9-s − 0.288·12-s + 1.10·13-s + 1/4·16-s − 0.458·19-s − 6/5·25-s − 0.192·27-s + 1.43·31-s + 1/6·36-s + 3.28·37-s − 0.640·39-s + 1.21·43-s − 0.144·48-s − 2·49-s + 0.554·52-s + 0.264·57-s + 3.58·61-s + 1/8·64-s − 2.93·67-s − 1.40·73-s + 0.692·75-s − 0.229·76-s − 0.900·79-s + 1/9·81-s − 0.829·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38988 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38988 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.275766994\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.275766994\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−10.56846810837709328705122029026, −9.614423289854631576305516246310, −9.600878448141019797397427731260, −8.666746132210989964286153997649, −7.987570621007900466899230715503, −7.87114970877127535090362209983, −6.93381933813641935760483898363, −6.46925034993373182658971924423, −5.88087715977795198376928556594, −5.66755609194805589043695166783, −4.39790054298019034493919320980, −4.27229588770101457625379469521, −3.17802027729570401593965112053, −2.33973473634575001863443152397, −1.15871174429474862980802238748,
1.15871174429474862980802238748, 2.33973473634575001863443152397, 3.17802027729570401593965112053, 4.27229588770101457625379469521, 4.39790054298019034493919320980, 5.66755609194805589043695166783, 5.88087715977795198376928556594, 6.46925034993373182658971924423, 6.93381933813641935760483898363, 7.87114970877127535090362209983, 7.987570621007900466899230715503, 8.666746132210989964286153997649, 9.600878448141019797397427731260, 9.614423289854631576305516246310, 10.56846810837709328705122029026
