| L(s) = 1 | + 0.463·3-s − 0.183·5-s + 7-s − 2.78·9-s + 4.93·11-s − 2.19·13-s − 0.0848·15-s + 17-s − 7.89·19-s + 0.463·21-s + 3.58·23-s − 4.96·25-s − 2.68·27-s − 2.53·29-s − 1.62·31-s + 2.28·33-s − 0.183·35-s − 9.29·37-s − 1.01·39-s − 0.851·41-s + 0.271·43-s + 0.509·45-s − 10.3·47-s + 49-s + 0.463·51-s + 13.0·53-s − 0.902·55-s + ⋯ |
| L(s) = 1 | + 0.267·3-s − 0.0818·5-s + 0.377·7-s − 0.928·9-s + 1.48·11-s − 0.609·13-s − 0.0219·15-s + 0.242·17-s − 1.81·19-s + 0.101·21-s + 0.746·23-s − 0.993·25-s − 0.516·27-s − 0.471·29-s − 0.291·31-s + 0.398·33-s − 0.0309·35-s − 1.52·37-s − 0.163·39-s − 0.132·41-s + 0.0414·43-s + 0.0759·45-s − 1.50·47-s + 0.142·49-s + 0.0649·51-s + 1.78·53-s − 0.121·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - 0.463T + 3T^{2} \) |
| 5 | \( 1 + 0.183T + 5T^{2} \) |
| 11 | \( 1 - 4.93T + 11T^{2} \) |
| 13 | \( 1 + 2.19T + 13T^{2} \) |
| 19 | \( 1 + 7.89T + 19T^{2} \) |
| 23 | \( 1 - 3.58T + 23T^{2} \) |
| 29 | \( 1 + 2.53T + 29T^{2} \) |
| 31 | \( 1 + 1.62T + 31T^{2} \) |
| 37 | \( 1 + 9.29T + 37T^{2} \) |
| 41 | \( 1 + 0.851T + 41T^{2} \) |
| 43 | \( 1 - 0.271T + 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 - 13.0T + 53T^{2} \) |
| 59 | \( 1 - 2.95T + 59T^{2} \) |
| 61 | \( 1 - 5.62T + 61T^{2} \) |
| 67 | \( 1 + 4.20T + 67T^{2} \) |
| 71 | \( 1 - 6.04T + 71T^{2} \) |
| 73 | \( 1 + 13.9T + 73T^{2} \) |
| 79 | \( 1 + 1.79T + 79T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 + 1.13T + 89T^{2} \) |
| 97 | \( 1 - 6.87T + 97T^{2} \) |
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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
−8.403804256208433685606302398244, −7.34956039979808044182779091284, −6.70330956014313623982826018323, −5.92081311536896610999949599694, −5.12908057576210545663525132330, −4.15658600913587284926085669748, −3.54568573677830076407064080664, −2.42113203266858542590335413159, −1.57955932164410891407881873258, 0,
1.57955932164410891407881873258, 2.42113203266858542590335413159, 3.54568573677830076407064080664, 4.15658600913587284926085669748, 5.12908057576210545663525132330, 5.92081311536896610999949599694, 6.70330956014313623982826018323, 7.34956039979808044182779091284, 8.403804256208433685606302398244
