| L(s) = 1 | − 2.75·2-s + 3-s + 5.58·4-s + 3.90·5-s − 2.75·6-s + 0.781·7-s − 9.86·8-s + 9-s − 10.7·10-s + 2.18·11-s + 5.58·12-s − 13-s − 2.15·14-s + 3.90·15-s + 16.0·16-s + 3.72·17-s − 2.75·18-s + 3.56·19-s + 21.7·20-s + 0.781·21-s − 6.01·22-s + 5.17·23-s − 9.86·24-s + 10.2·25-s + 2.75·26-s + 27-s + 4.36·28-s + ⋯ |
| L(s) = 1 | − 1.94·2-s + 0.577·3-s + 2.79·4-s + 1.74·5-s − 1.12·6-s + 0.295·7-s − 3.48·8-s + 0.333·9-s − 3.39·10-s + 0.658·11-s + 1.61·12-s − 0.277·13-s − 0.574·14-s + 1.00·15-s + 4.00·16-s + 0.903·17-s − 0.649·18-s + 0.818·19-s + 4.87·20-s + 0.170·21-s − 1.28·22-s + 1.07·23-s − 2.01·24-s + 2.04·25-s + 0.540·26-s + 0.192·27-s + 0.824·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.779970364\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.779970364\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 + 2.75T + 2T^{2} \) |
| 5 | \( 1 - 3.90T + 5T^{2} \) |
| 7 | \( 1 - 0.781T + 7T^{2} \) |
| 11 | \( 1 - 2.18T + 11T^{2} \) |
| 17 | \( 1 - 3.72T + 17T^{2} \) |
| 19 | \( 1 - 3.56T + 19T^{2} \) |
| 23 | \( 1 - 5.17T + 23T^{2} \) |
| 29 | \( 1 - 4.04T + 29T^{2} \) |
| 31 | \( 1 - 6.84T + 31T^{2} \) |
| 37 | \( 1 - 8.92T + 37T^{2} \) |
| 41 | \( 1 + 10.7T + 41T^{2} \) |
| 43 | \( 1 + 2.19T + 43T^{2} \) |
| 47 | \( 1 + 4.77T + 47T^{2} \) |
| 53 | \( 1 + 0.781T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 - 1.10T + 61T^{2} \) |
| 67 | \( 1 - 12.5T + 67T^{2} \) |
| 71 | \( 1 + 8.75T + 71T^{2} \) |
| 73 | \( 1 - 6.59T + 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 + 7.23T + 83T^{2} \) |
| 89 | \( 1 + 12.5T + 89T^{2} \) |
| 97 | \( 1 - 1.94T + 97T^{2} \) |
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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
−8.511814904203294625205927949718, −8.057551175406384422561436341843, −7.06808179356169812084752448035, −6.58985367279048518249031259349, −5.85491085661557597771881813689, −4.94655877078948265651500094203, −3.11475574726284747453150320714, −2.63875608124186620293149118415, −1.52850103867190951829026661555, −1.17156229289525806220111805614,
1.17156229289525806220111805614, 1.52850103867190951829026661555, 2.63875608124186620293149118415, 3.11475574726284747453150320714, 4.94655877078948265651500094203, 5.85491085661557597771881813689, 6.58985367279048518249031259349, 7.06808179356169812084752448035, 8.057551175406384422561436341843, 8.511814904203294625205927949718
