| L(s) = 1 | − 3-s − 5-s + 0.259·7-s + 9-s + 4·11-s + 2.25·13-s + 15-s + 3.31·17-s + 6.10·19-s − 0.259·21-s − 0.791·23-s + 25-s − 27-s − 2.53·29-s − 31-s − 4·33-s − 0.259·35-s − 4.36·37-s − 2.25·39-s + 10.2·41-s − 10.1·43-s − 45-s − 7.41·47-s − 6.93·49-s − 3.31·51-s − 0.689·53-s − 4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.0980·7-s + 0.333·9-s + 1.20·11-s + 0.626·13-s + 0.258·15-s + 0.803·17-s + 1.40·19-s − 0.0566·21-s − 0.165·23-s + 0.200·25-s − 0.192·27-s − 0.470·29-s − 0.179·31-s − 0.696·33-s − 0.0438·35-s − 0.717·37-s − 0.361·39-s + 1.59·41-s − 1.54·43-s − 0.149·45-s − 1.08·47-s − 0.990·49-s − 0.463·51-s − 0.0946·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.672565495\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.672565495\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 - 0.259T + 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 - 2.25T + 13T^{2} \) |
| 17 | \( 1 - 3.31T + 17T^{2} \) |
| 19 | \( 1 - 6.10T + 19T^{2} \) |
| 23 | \( 1 + 0.791T + 23T^{2} \) |
| 29 | \( 1 + 2.53T + 29T^{2} \) |
| 37 | \( 1 + 4.36T + 37T^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 + 7.41T + 47T^{2} \) |
| 53 | \( 1 + 0.689T + 53T^{2} \) |
| 59 | \( 1 - 5.05T + 59T^{2} \) |
| 61 | \( 1 + 1.48T + 61T^{2} \) |
| 67 | \( 1 - 5.84T + 67T^{2} \) |
| 71 | \( 1 - 9.05T + 71T^{2} \) |
| 73 | \( 1 - 0.362T + 73T^{2} \) |
| 79 | \( 1 - 4.79T + 79T^{2} \) |
| 83 | \( 1 - 18.0T + 83T^{2} \) |
| 89 | \( 1 + 9.15T + 89T^{2} \) |
| 97 | \( 1 - 9.68T + 97T^{2} \) |
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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
−8.423839374931007248616549953960, −7.76285630734243294717627489192, −6.97764726237761041687414354226, −6.33009998171823856492027622091, −5.51599317071684454436331871965, −4.80640147779905441950500525545, −3.76986000468972610681989756404, −3.31270439743374466248981479618, −1.71281293109124348627785315021, −0.831369387683166627415918848051,
0.831369387683166627415918848051, 1.71281293109124348627785315021, 3.31270439743374466248981479618, 3.76986000468972610681989756404, 4.80640147779905441950500525545, 5.51599317071684454436331871965, 6.33009998171823856492027622091, 6.97764726237761041687414354226, 7.76285630734243294717627489192, 8.423839374931007248616549953960
