| L(s) = 1 | − 2.40·2-s − 2.46·3-s + 3.77·4-s + 5.93·6-s − 0.299·7-s − 4.25·8-s + 3.09·9-s + 5.75·11-s − 9.31·12-s + 6.54·13-s + 0.719·14-s + 2.68·16-s + 2.71·17-s − 7.43·18-s − 0.424·19-s + 0.738·21-s − 13.8·22-s + 7.86·23-s + 10.5·24-s − 15.7·26-s − 0.232·27-s − 1.12·28-s + 2.33·29-s + 4.98·31-s + 2.06·32-s − 14.1·33-s − 6.51·34-s + ⋯ |
| L(s) = 1 | − 1.69·2-s − 1.42·3-s + 1.88·4-s + 2.42·6-s − 0.113·7-s − 1.50·8-s + 1.03·9-s + 1.73·11-s − 2.68·12-s + 1.81·13-s + 0.192·14-s + 0.671·16-s + 0.658·17-s − 1.75·18-s − 0.0974·19-s + 0.161·21-s − 2.94·22-s + 1.64·23-s + 2.14·24-s − 3.08·26-s − 0.0447·27-s − 0.213·28-s + 0.432·29-s + 0.895·31-s + 0.364·32-s − 2.47·33-s − 1.11·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5975698624\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5975698624\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 + 2.40T + 2T^{2} \) |
| 3 | \( 1 + 2.46T + 3T^{2} \) |
| 7 | \( 1 + 0.299T + 7T^{2} \) |
| 11 | \( 1 - 5.75T + 11T^{2} \) |
| 13 | \( 1 - 6.54T + 13T^{2} \) |
| 17 | \( 1 - 2.71T + 17T^{2} \) |
| 19 | \( 1 + 0.424T + 19T^{2} \) |
| 23 | \( 1 - 7.86T + 23T^{2} \) |
| 29 | \( 1 - 2.33T + 29T^{2} \) |
| 31 | \( 1 - 4.98T + 31T^{2} \) |
| 37 | \( 1 - 3.99T + 37T^{2} \) |
| 41 | \( 1 + 0.478T + 41T^{2} \) |
| 43 | \( 1 + 7.57T + 43T^{2} \) |
| 47 | \( 1 - 6.17T + 47T^{2} \) |
| 53 | \( 1 + 9.28T + 53T^{2} \) |
| 59 | \( 1 - 7.23T + 59T^{2} \) |
| 67 | \( 1 + 2.04T + 67T^{2} \) |
| 71 | \( 1 - 1.85T + 71T^{2} \) |
| 73 | \( 1 + 8.43T + 73T^{2} \) |
| 79 | \( 1 - 6.12T + 79T^{2} \) |
| 83 | \( 1 - 2.19T + 83T^{2} \) |
| 89 | \( 1 - 5.03T + 89T^{2} \) |
| 97 | \( 1 + 18.1T + 97T^{2} \) |
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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
−9.418426566243479416575552429402, −8.804607233385418200106312542309, −8.070061960619397569941472745369, −6.79859484060578777624600555098, −6.56222302212609681519528680742, −5.80451318049209403991055655831, −4.52978996541550039749694446574, −3.28190990021048240957131169297, −1.37696177681412274718410402571, −0.908550211616597303218331259333,
0.908550211616597303218331259333, 1.37696177681412274718410402571, 3.28190990021048240957131169297, 4.52978996541550039749694446574, 5.80451318049209403991055655831, 6.56222302212609681519528680742, 6.79859484060578777624600555098, 8.070061960619397569941472745369, 8.804607233385418200106312542309, 9.418426566243479416575552429402
