| L(s) = 1 | + 0.800·2-s − 1.35·4-s + 0.757·5-s + 0.484·7-s − 2.68·8-s + 0.606·10-s − 11-s − 5.01·13-s + 0.388·14-s + 0.564·16-s − 0.818·17-s + 1.42·19-s − 1.02·20-s − 0.800·22-s + 2.31·23-s − 4.42·25-s − 4.01·26-s − 0.658·28-s − 4.07·29-s + 8.39·31-s + 5.83·32-s − 0.655·34-s + 0.367·35-s − 5.04·37-s + 1.14·38-s − 2.03·40-s − 4.54·41-s + ⋯ |
| L(s) = 1 | + 0.566·2-s − 0.679·4-s + 0.338·5-s + 0.183·7-s − 0.950·8-s + 0.191·10-s − 0.301·11-s − 1.38·13-s + 0.103·14-s + 0.141·16-s − 0.198·17-s + 0.326·19-s − 0.230·20-s − 0.170·22-s + 0.482·23-s − 0.885·25-s − 0.786·26-s − 0.124·28-s − 0.756·29-s + 1.50·31-s + 1.03·32-s − 0.112·34-s + 0.0620·35-s − 0.828·37-s + 0.185·38-s − 0.322·40-s − 0.710·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.668980541\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.668980541\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 - 0.800T + 2T^{2} \) |
| 5 | \( 1 - 0.757T + 5T^{2} \) |
| 7 | \( 1 - 0.484T + 7T^{2} \) |
| 13 | \( 1 + 5.01T + 13T^{2} \) |
| 17 | \( 1 + 0.818T + 17T^{2} \) |
| 19 | \( 1 - 1.42T + 19T^{2} \) |
| 23 | \( 1 - 2.31T + 23T^{2} \) |
| 29 | \( 1 + 4.07T + 29T^{2} \) |
| 31 | \( 1 - 8.39T + 31T^{2} \) |
| 37 | \( 1 + 5.04T + 37T^{2} \) |
| 41 | \( 1 + 4.54T + 41T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 - 0.677T + 47T^{2} \) |
| 53 | \( 1 - 3.48T + 53T^{2} \) |
| 59 | \( 1 - 13.2T + 59T^{2} \) |
| 67 | \( 1 + 1.60T + 67T^{2} \) |
| 71 | \( 1 + 3.46T + 71T^{2} \) |
| 73 | \( 1 - 9.56T + 73T^{2} \) |
| 79 | \( 1 - 2.71T + 79T^{2} \) |
| 83 | \( 1 - 1.98T + 83T^{2} \) |
| 89 | \( 1 + 0.245T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−8.051838658308311566993954477521, −7.37856530404260533063632161847, −6.56046522254144669866437785370, −5.69498556970871896353229610450, −5.14446471789457428612891773374, −4.57901314913007551887447382426, −3.75111351036982102475272807952, −2.84966776767163455300813814906, −2.06042251172998043442841036415, −0.60624782593070724450386777892,
0.60624782593070724450386777892, 2.06042251172998043442841036415, 2.84966776767163455300813814906, 3.75111351036982102475272807952, 4.57901314913007551887447382426, 5.14446471789457428612891773374, 5.69498556970871896353229610450, 6.56046522254144669866437785370, 7.37856530404260533063632161847, 8.051838658308311566993954477521
