| L(s) = 1 | + 2.93·5-s + 4.68·7-s − 0.762·11-s + 1.76·13-s − 3.36·17-s + 7.36·19-s − 3.25·23-s + 3.61·25-s + 3.25·29-s + 3.06·31-s + 13.7·35-s − 37-s + 7.42·41-s − 12.2·43-s − 0.302·47-s + 14.9·49-s − 5.53·53-s − 2.23·55-s − 10.2·59-s + 12.2·61-s + 5.17·65-s − 13.1·67-s − 0.173·71-s − 1.23·73-s − 3.56·77-s + 4.61·79-s − 3.53·83-s + ⋯ |
| L(s) = 1 | + 1.31·5-s + 1.76·7-s − 0.229·11-s + 0.488·13-s − 0.815·17-s + 1.68·19-s − 0.678·23-s + 0.723·25-s + 0.604·29-s + 0.550·31-s + 2.32·35-s − 0.164·37-s + 1.15·41-s − 1.86·43-s − 0.0440·47-s + 2.13·49-s − 0.760·53-s − 0.301·55-s − 1.33·59-s + 1.57·61-s + 0.641·65-s − 1.60·67-s − 0.0205·71-s − 0.144·73-s − 0.406·77-s + 0.519·79-s − 0.388·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.044797482\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.044797482\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 5 | \( 1 - 2.93T + 5T^{2} \) |
| 7 | \( 1 - 4.68T + 7T^{2} \) |
| 11 | \( 1 + 0.762T + 11T^{2} \) |
| 13 | \( 1 - 1.76T + 13T^{2} \) |
| 17 | \( 1 + 3.36T + 17T^{2} \) |
| 19 | \( 1 - 7.36T + 19T^{2} \) |
| 23 | \( 1 + 3.25T + 23T^{2} \) |
| 29 | \( 1 - 3.25T + 29T^{2} \) |
| 31 | \( 1 - 3.06T + 31T^{2} \) |
| 41 | \( 1 - 7.42T + 41T^{2} \) |
| 43 | \( 1 + 12.2T + 43T^{2} \) |
| 47 | \( 1 + 0.302T + 47T^{2} \) |
| 53 | \( 1 + 5.53T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 + 0.173T + 71T^{2} \) |
| 73 | \( 1 + 1.23T + 73T^{2} \) |
| 79 | \( 1 - 4.61T + 79T^{2} \) |
| 83 | \( 1 + 3.53T + 83T^{2} \) |
| 89 | \( 1 + 15.7T + 89T^{2} \) |
| 97 | \( 1 + 16.1T + 97T^{2} \) |
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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
−8.786328045990009409336192536252, −8.164445466033670956678350420521, −7.43460973313759157576188468306, −6.44421415026960976901966693936, −5.63952546765489272486691049621, −5.05513544279625830352625172154, −4.30859222905583820755562710161, −2.92995999126187042135093269501, −1.92415209647921171034781719101, −1.26417473221193574661965776433,
1.26417473221193574661965776433, 1.92415209647921171034781719101, 2.92995999126187042135093269501, 4.30859222905583820755562710161, 5.05513544279625830352625172154, 5.63952546765489272486691049621, 6.44421415026960976901966693936, 7.43460973313759157576188468306, 8.164445466033670956678350420521, 8.786328045990009409336192536252
