| L(s) = 1 | − 3-s + 4·7-s + 9-s − 11-s − 5·13-s + 3·17-s − 4·21-s + 3·23-s − 27-s − 8·29-s + 31-s + 33-s − 8·37-s + 5·39-s − 41-s − 43-s + 9·49-s − 3·51-s − 3·53-s + 8·61-s + 4·63-s − 9·67-s − 3·69-s − 4·71-s − 4·73-s − 4·77-s − 8·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.51·7-s + 1/3·9-s − 0.301·11-s − 1.38·13-s + 0.727·17-s − 0.872·21-s + 0.625·23-s − 0.192·27-s − 1.48·29-s + 0.179·31-s + 0.174·33-s − 1.31·37-s + 0.800·39-s − 0.156·41-s − 0.152·43-s + 9/7·49-s − 0.420·51-s − 0.412·53-s + 1.02·61-s + 0.503·63-s − 1.09·67-s − 0.361·69-s − 0.474·71-s − 0.468·73-s − 0.455·77-s − 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.609792604\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.609792604\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 - 15 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.40864891023650, −14.25953157521378, −13.37557131950819, −12.92341261415964, −12.32967275553776, −11.78616036518228, −11.55705784998953, −10.90233777397990, −10.41373395764156, −9.989232772012115, −9.282056085456019, −8.752488396173857, −8.060760036873995, −7.524814728680606, −7.288914887499493, −6.564221497561844, −5.643004343585459, −5.315220361455454, −4.871561143574158, −4.341191801037110, −3.537314217433900, −2.737793951256937, −1.929191330552354, −1.468745068084233, −0.4595361651373856,
0.4595361651373856, 1.468745068084233, 1.929191330552354, 2.737793951256937, 3.537314217433900, 4.341191801037110, 4.871561143574158, 5.315220361455454, 5.643004343585459, 6.564221497561844, 7.288914887499493, 7.524814728680606, 8.060760036873995, 8.752488396173857, 9.282056085456019, 9.989232772012115, 10.41373395764156, 10.90233777397990, 11.55705784998953, 11.78616036518228, 12.32967275553776, 12.92341261415964, 13.37557131950819, 14.25953157521378, 14.40864891023650
