| L(s) = 1 | − 0.365·3-s + 3.51·5-s − 1.20·7-s − 2.86·9-s − 11-s + 1.20·13-s − 1.28·15-s − 3.15·17-s − 19-s + 0.439·21-s − 6.54·23-s + 7.39·25-s + 2.14·27-s + 4.07·29-s − 5.34·31-s + 0.365·33-s − 4.23·35-s + 1.39·37-s − 0.439·39-s − 0.683·41-s − 3.62·43-s − 10.0·45-s − 5.67·47-s − 5.55·49-s + 1.15·51-s − 0.523·53-s − 3.51·55-s + ⋯ |
| L(s) = 1 | − 0.211·3-s + 1.57·5-s − 0.454·7-s − 0.955·9-s − 0.301·11-s + 0.333·13-s − 0.332·15-s − 0.764·17-s − 0.229·19-s + 0.0959·21-s − 1.36·23-s + 1.47·25-s + 0.413·27-s + 0.756·29-s − 0.960·31-s + 0.0636·33-s − 0.715·35-s + 0.229·37-s − 0.0704·39-s − 0.106·41-s − 0.552·43-s − 1.50·45-s − 0.827·47-s − 0.793·49-s + 0.161·51-s − 0.0719·53-s − 0.474·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 0.365T + 3T^{2} \) |
| 5 | \( 1 - 3.51T + 5T^{2} \) |
| 7 | \( 1 + 1.20T + 7T^{2} \) |
| 13 | \( 1 - 1.20T + 13T^{2} \) |
| 17 | \( 1 + 3.15T + 17T^{2} \) |
| 23 | \( 1 + 6.54T + 23T^{2} \) |
| 29 | \( 1 - 4.07T + 29T^{2} \) |
| 31 | \( 1 + 5.34T + 31T^{2} \) |
| 37 | \( 1 - 1.39T + 37T^{2} \) |
| 41 | \( 1 + 0.683T + 41T^{2} \) |
| 43 | \( 1 + 3.62T + 43T^{2} \) |
| 47 | \( 1 + 5.67T + 47T^{2} \) |
| 53 | \( 1 + 0.523T + 53T^{2} \) |
| 59 | \( 1 - 8.37T + 59T^{2} \) |
| 61 | \( 1 + 6.55T + 61T^{2} \) |
| 67 | \( 1 + 1.71T + 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 + 14.5T + 73T^{2} \) |
| 79 | \( 1 + 10.6T + 79T^{2} \) |
| 83 | \( 1 - 7.55T + 83T^{2} \) |
| 89 | \( 1 + 10.6T + 89T^{2} \) |
| 97 | \( 1 - 12.8T + 97T^{2} \) |
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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
−8.565280489947777239847418207387, −7.43403966337527409225580332508, −6.32590293117036371546393022239, −6.15131377238035267492971002334, −5.40372046973107968626149287422, −4.54602604993324896542166707725, −3.30086376738166310562073301557, −2.45967403630915437610332378810, −1.64989814659290967461581568727, 0,
1.64989814659290967461581568727, 2.45967403630915437610332378810, 3.30086376738166310562073301557, 4.54602604993324896542166707725, 5.40372046973107968626149287422, 6.15131377238035267492971002334, 6.32590293117036371546393022239, 7.43403966337527409225580332508, 8.565280489947777239847418207387
