| L(s) = 1 | + 3-s − 7-s + 9-s + 2.68·11-s − 6.97·13-s + 2.29·17-s − 19-s − 21-s − 4.39·23-s − 5·25-s + 27-s + 0.978·29-s − 7.66·31-s + 2.68·33-s + 5.66·37-s − 6.97·39-s + 7.37·41-s − 9.37·43-s − 4.68·47-s + 49-s + 2.29·51-s − 12.9·53-s − 57-s − 0.585·59-s − 6.58·61-s − 63-s + 12.0·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 0.333·9-s + 0.809·11-s − 1.93·13-s + 0.556·17-s − 0.229·19-s − 0.218·21-s − 0.916·23-s − 25-s + 0.192·27-s + 0.181·29-s − 1.37·31-s + 0.467·33-s + 0.931·37-s − 1.11·39-s + 1.15·41-s − 1.42·43-s − 0.683·47-s + 0.142·49-s + 0.321·51-s − 1.78·53-s − 0.132·57-s − 0.0762·59-s − 0.843·61-s − 0.125·63-s + 1.47·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 - 2.68T + 11T^{2} \) |
| 13 | \( 1 + 6.97T + 13T^{2} \) |
| 17 | \( 1 - 2.29T + 17T^{2} \) |
| 23 | \( 1 + 4.39T + 23T^{2} \) |
| 29 | \( 1 - 0.978T + 29T^{2} \) |
| 31 | \( 1 + 7.66T + 31T^{2} \) |
| 37 | \( 1 - 5.66T + 37T^{2} \) |
| 41 | \( 1 - 7.37T + 41T^{2} \) |
| 43 | \( 1 + 9.37T + 43T^{2} \) |
| 47 | \( 1 + 4.68T + 47T^{2} \) |
| 53 | \( 1 + 12.9T + 53T^{2} \) |
| 59 | \( 1 + 0.585T + 59T^{2} \) |
| 61 | \( 1 + 6.58T + 61T^{2} \) |
| 67 | \( 1 - 12.0T + 67T^{2} \) |
| 71 | \( 1 + 1.60T + 71T^{2} \) |
| 73 | \( 1 + 7.37T + 73T^{2} \) |
| 79 | \( 1 - 14.3T + 79T^{2} \) |
| 83 | \( 1 - 8.35T + 83T^{2} \) |
| 89 | \( 1 + 15.9T + 89T^{2} \) |
| 97 | \( 1 - 5.27T + 97T^{2} \) |
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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
−8.109445063262521776567204048584, −7.68378547798668239242355589224, −6.88293328983520727975241193468, −6.11924444577045419169008902863, −5.16266868759834797226565001717, −4.28957533020980283845120205006, −3.51266154468211163360502961187, −2.55503576553075487499078067936, −1.68510107726780480065162681910, 0,
1.68510107726780480065162681910, 2.55503576553075487499078067936, 3.51266154468211163360502961187, 4.28957533020980283845120205006, 5.16266868759834797226565001717, 6.11924444577045419169008902863, 6.88293328983520727975241193468, 7.68378547798668239242355589224, 8.109445063262521776567204048584
