| L(s) = 1 | − 3-s − 0.732·5-s − 2·7-s + 9-s + 1.46·11-s + 3.46·13-s + 0.732·15-s + 0.732·17-s − 5.46·19-s + 2·21-s − 1.26·23-s − 4.46·25-s − 27-s − 6.19·29-s − 4·31-s − 1.46·33-s + 1.46·35-s − 37-s − 3.46·39-s − 2·41-s − 10.9·43-s − 0.732·45-s − 6.92·47-s − 3·49-s − 0.732·51-s + 10.3·53-s − 1.07·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.327·5-s − 0.755·7-s + 0.333·9-s + 0.441·11-s + 0.960·13-s + 0.189·15-s + 0.177·17-s − 1.25·19-s + 0.436·21-s − 0.264·23-s − 0.892·25-s − 0.192·27-s − 1.15·29-s − 0.718·31-s − 0.254·33-s + 0.247·35-s − 0.164·37-s − 0.554·39-s − 0.312·41-s − 1.66·43-s − 0.109·45-s − 1.01·47-s − 0.428·49-s − 0.102·51-s + 1.42·53-s − 0.144·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| good | 5 | \( 1 + 0.732T + 5T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 - 1.46T + 11T^{2} \) |
| 13 | \( 1 - 3.46T + 13T^{2} \) |
| 17 | \( 1 - 0.732T + 17T^{2} \) |
| 19 | \( 1 + 5.46T + 19T^{2} \) |
| 23 | \( 1 + 1.26T + 23T^{2} \) |
| 29 | \( 1 + 6.19T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 - 1.26T + 59T^{2} \) |
| 61 | \( 1 - 4.92T + 61T^{2} \) |
| 67 | \( 1 + 6T + 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 + 5.46T + 73T^{2} \) |
| 79 | \( 1 + 2.53T + 79T^{2} \) |
| 83 | \( 1 + 2.53T + 83T^{2} \) |
| 89 | \( 1 + 2.19T + 89T^{2} \) |
| 97 | \( 1 - 2.39T + 97T^{2} \) |
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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
−9.804500009126266480529235270676, −8.894090084411864868085469021266, −8.048593573838398781703861585157, −6.93051017174407332581792160104, −6.28216157537238685306256902433, −5.45620909190558287087790382577, −4.13976214866306107342599690879, −3.46709519574716159658348823481, −1.77326906115428539823514408004, 0,
1.77326906115428539823514408004, 3.46709519574716159658348823481, 4.13976214866306107342599690879, 5.45620909190558287087790382577, 6.28216157537238685306256902433, 6.93051017174407332581792160104, 8.048593573838398781703861585157, 8.894090084411864868085469021266, 9.804500009126266480529235270676
