| L(s) = 1 | + 0.152·2-s + 3-s − 1.97·4-s + 0.152·6-s + 2.73·7-s − 0.606·8-s + 9-s − 0.976·11-s − 1.97·12-s + 2.15·13-s + 0.417·14-s + 3.86·16-s + 5.31·17-s + 0.152·18-s + 7.23·19-s + 2.73·21-s − 0.148·22-s − 5.21·23-s − 0.606·24-s + 0.328·26-s + 27-s − 5.40·28-s + 5·29-s − 10.0·31-s + 1.80·32-s − 0.976·33-s + 0.810·34-s + ⋯ |
| L(s) = 1 | + 0.107·2-s + 0.577·3-s − 0.988·4-s + 0.0622·6-s + 1.03·7-s − 0.214·8-s + 0.333·9-s − 0.294·11-s − 0.570·12-s + 0.596·13-s + 0.111·14-s + 0.965·16-s + 1.28·17-s + 0.0359·18-s + 1.65·19-s + 0.596·21-s − 0.0317·22-s − 1.08·23-s − 0.123·24-s + 0.0643·26-s + 0.192·27-s − 1.02·28-s + 0.928·29-s − 1.80·31-s + 0.318·32-s − 0.170·33-s + 0.139·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.386405998\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.386405998\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 - 0.152T + 2T^{2} \) |
| 7 | \( 1 - 2.73T + 7T^{2} \) |
| 11 | \( 1 + 0.976T + 11T^{2} \) |
| 13 | \( 1 - 2.15T + 13T^{2} \) |
| 17 | \( 1 - 5.31T + 17T^{2} \) |
| 19 | \( 1 - 7.23T + 19T^{2} \) |
| 23 | \( 1 + 5.21T + 23T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 + 5.83T + 37T^{2} \) |
| 41 | \( 1 - 7.97T + 41T^{2} \) |
| 43 | \( 1 + 9.42T + 43T^{2} \) |
| 53 | \( 1 + 8.86T + 53T^{2} \) |
| 59 | \( 1 - 5.88T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 - 14.3T + 71T^{2} \) |
| 73 | \( 1 - 12.4T + 73T^{2} \) |
| 79 | \( 1 - 14.7T + 79T^{2} \) |
| 83 | \( 1 - 9.03T + 83T^{2} \) |
| 89 | \( 1 - 6.78T + 89T^{2} \) |
| 97 | \( 1 - 6.55T + 97T^{2} \) |
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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
−8.418194134754333268469890330706, −7.941072810726347769505024502465, −7.49145267851032571036562422244, −6.19486724658767724019178576193, −5.20518850101923157110084351111, −4.97174111206264922356876369587, −3.68922822501375424007833532641, −3.37810780552581810998870078853, −1.90959044423804297946054798185, −0.940137677838597890420795055465,
0.940137677838597890420795055465, 1.90959044423804297946054798185, 3.37810780552581810998870078853, 3.68922822501375424007833532641, 4.97174111206264922356876369587, 5.20518850101923157110084351111, 6.19486724658767724019178576193, 7.49145267851032571036562422244, 7.941072810726347769505024502465, 8.418194134754333268469890330706
