| L(s) = 1 | − 3-s − 4·7-s + 9-s + 5·11-s + 3·13-s − 17-s + 4·19-s + 4·21-s − 7·23-s − 27-s + 7·31-s − 5·33-s + 4·37-s − 3·39-s − 9·41-s + 43-s + 9·49-s + 51-s − 3·53-s − 4·57-s + 8·59-s − 4·61-s − 4·63-s + 67-s + 7·69-s + 12·71-s − 20·77-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s + 1.50·11-s + 0.832·13-s − 0.242·17-s + 0.917·19-s + 0.872·21-s − 1.45·23-s − 0.192·27-s + 1.25·31-s − 0.870·33-s + 0.657·37-s − 0.480·39-s − 1.40·41-s + 0.152·43-s + 9/7·49-s + 0.140·51-s − 0.412·53-s − 0.529·57-s + 1.04·59-s − 0.512·61-s − 0.503·63-s + 0.122·67-s + 0.842·69-s + 1.42·71-s − 2.27·77-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
| \(L(1)\) |
\(\approx\) |
\(1.657268759\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.657268759\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−14.36448046729030, −13.83760677743243, −13.51652452078853, −12.96891547715750, −12.23277133076480, −12.03059593442319, −11.51689513483478, −10.95664387024862, −10.23262017579600, −9.729796020782137, −9.540427191510648, −8.812872487634597, −8.266159714918286, −7.552430149418869, −6.742065859425639, −6.491068897801101, −6.130103491089743, −5.516915897760608, −4.693922254950877, −3.932040636425093, −3.647012212360758, −2.978513175380696, −2.036369502871696, −1.188443852775662, −0.5238468654081729,
0.5238468654081729, 1.188443852775662, 2.036369502871696, 2.978513175380696, 3.647012212360758, 3.932040636425093, 4.693922254950877, 5.516915897760608, 6.130103491089743, 6.491068897801101, 6.742065859425639, 7.552430149418869, 8.266159714918286, 8.812872487634597, 9.540427191510648, 9.729796020782137, 10.23262017579600, 10.95664387024862, 11.51689513483478, 12.03059593442319, 12.23277133076480, 12.96891547715750, 13.51652452078853, 13.83760677743243, 14.36448046729030
