| L(s) = 1 | + 1.79·3-s + 3.79·5-s − 7-s + 0.208·9-s + 3.79·11-s + 6.79·15-s + 3·17-s − 7.37·19-s − 1.79·21-s + 6·23-s + 9.37·25-s − 5.00·27-s − 2.20·29-s + 31-s + 6.79·33-s − 3.79·35-s + 4·37-s + 1.58·41-s + 4.37·43-s + 0.791·45-s + 12.1·47-s + 49-s + 5.37·51-s − 10.5·53-s + 14.3·55-s − 13.2·57-s + 9·59-s + ⋯ |
| L(s) = 1 | + 1.03·3-s + 1.69·5-s − 0.377·7-s + 0.0695·9-s + 1.14·11-s + 1.75·15-s + 0.727·17-s − 1.69·19-s − 0.390·21-s + 1.25·23-s + 1.87·25-s − 0.962·27-s − 0.410·29-s + 0.179·31-s + 1.18·33-s − 0.640·35-s + 0.657·37-s + 0.247·41-s + 0.667·43-s + 0.117·45-s + 1.77·47-s + 0.142·49-s + 0.752·51-s − 1.45·53-s + 1.93·55-s − 1.74·57-s + 1.17·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4732 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4732 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.095230348\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.095230348\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 1.79T + 3T^{2} \) |
| 5 | \( 1 - 3.79T + 5T^{2} \) |
| 11 | \( 1 - 3.79T + 11T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 + 7.37T + 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 + 2.20T + 29T^{2} \) |
| 31 | \( 1 - T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 - 1.58T + 41T^{2} \) |
| 43 | \( 1 - 4.37T + 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 - 9T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 7T + 67T^{2} \) |
| 71 | \( 1 + 7.58T + 71T^{2} \) |
| 73 | \( 1 + 14T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 9T + 83T^{2} \) |
| 89 | \( 1 - 5.37T + 89T^{2} \) |
| 97 | \( 1 - 18.3T + 97T^{2} \) |
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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
−8.632164310327843624409446583232, −7.59341398981278110193384535627, −6.74100028685390125904658899133, −6.12551863375761642811233815104, −5.58224591100784967254643811304, −4.48098865359157370573716348482, −3.59305087609701513458108466387, −2.69726886461902622718065349987, −2.11474084394772602895127858407, −1.14728394116008939265892258862,
1.14728394116008939265892258862, 2.11474084394772602895127858407, 2.69726886461902622718065349987, 3.59305087609701513458108466387, 4.48098865359157370573716348482, 5.58224591100784967254643811304, 6.12551863375761642811233815104, 6.74100028685390125904658899133, 7.59341398981278110193384535627, 8.632164310327843624409446583232
