| L(s) = 1 | + 1.76·3-s + 0.469·5-s − 1.03·7-s + 0.123·9-s + 1.84·11-s − 0.700·13-s + 0.830·15-s + 0.793·17-s − 3.65·19-s − 1.83·21-s + 3.65·23-s − 4.77·25-s − 5.08·27-s + 5.52·29-s − 6.13·31-s + 3.25·33-s − 0.487·35-s − 2.73·37-s − 1.23·39-s − 9.69·41-s + 4.25·43-s + 0.0580·45-s + 5.21·47-s − 5.92·49-s + 1.40·51-s − 7.01·53-s + 0.865·55-s + ⋯ |
| L(s) = 1 | + 1.02·3-s + 0.210·5-s − 0.392·7-s + 0.0411·9-s + 0.555·11-s − 0.194·13-s + 0.214·15-s + 0.192·17-s − 0.839·19-s − 0.400·21-s + 0.761·23-s − 0.955·25-s − 0.978·27-s + 1.02·29-s − 1.10·31-s + 0.566·33-s − 0.0824·35-s − 0.448·37-s − 0.198·39-s − 1.51·41-s + 0.648·43-s + 0.00864·45-s + 0.760·47-s − 0.845·49-s + 0.196·51-s − 0.963·53-s + 0.116·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8752 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8752 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 547 | \( 1 - T \) |
| good | 3 | \( 1 - 1.76T + 3T^{2} \) |
| 5 | \( 1 - 0.469T + 5T^{2} \) |
| 7 | \( 1 + 1.03T + 7T^{2} \) |
| 11 | \( 1 - 1.84T + 11T^{2} \) |
| 13 | \( 1 + 0.700T + 13T^{2} \) |
| 17 | \( 1 - 0.793T + 17T^{2} \) |
| 19 | \( 1 + 3.65T + 19T^{2} \) |
| 23 | \( 1 - 3.65T + 23T^{2} \) |
| 29 | \( 1 - 5.52T + 29T^{2} \) |
| 31 | \( 1 + 6.13T + 31T^{2} \) |
| 37 | \( 1 + 2.73T + 37T^{2} \) |
| 41 | \( 1 + 9.69T + 41T^{2} \) |
| 43 | \( 1 - 4.25T + 43T^{2} \) |
| 47 | \( 1 - 5.21T + 47T^{2} \) |
| 53 | \( 1 + 7.01T + 53T^{2} \) |
| 59 | \( 1 + 6.98T + 59T^{2} \) |
| 61 | \( 1 + 3.87T + 61T^{2} \) |
| 67 | \( 1 - 5.10T + 67T^{2} \) |
| 71 | \( 1 + 1.33T + 71T^{2} \) |
| 73 | \( 1 + 7.83T + 73T^{2} \) |
| 79 | \( 1 + 4.36T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 + 0.208T + 89T^{2} \) |
| 97 | \( 1 - 3.08T + 97T^{2} \) |
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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
−7.54941045434904832238529498718, −6.73952946153967430966177942261, −6.17026152073813472841034549364, −5.35228438287458947934769424269, −4.48600566572029746593693264950, −3.65185809604334886061582605654, −3.11179699013185755992282942388, −2.27469097957591824803039110757, −1.49807641431596409880722205230, 0,
1.49807641431596409880722205230, 2.27469097957591824803039110757, 3.11179699013185755992282942388, 3.65185809604334886061582605654, 4.48600566572029746593693264950, 5.35228438287458947934769424269, 6.17026152073813472841034549364, 6.73952946153967430966177942261, 7.54941045434904832238529498718
