| L(s) = 1 | + 0.381·3-s + 3·7-s − 2.85·9-s + 3·11-s − 4.85·13-s + 4.23·17-s + 3.61·19-s + 1.14·21-s + 1.23·23-s − 2.23·27-s + 6.70·29-s − 5.09·31-s + 1.14·33-s + 3.70·37-s − 1.85·39-s − 3·41-s + 9·43-s + 8.32·47-s + 2·49-s + 1.61·51-s + 4.61·53-s + 1.38·57-s − 4.14·59-s − 6.09·61-s − 8.56·63-s + 13.8·67-s + 0.472·69-s + ⋯ |
| L(s) = 1 | + 0.220·3-s + 1.13·7-s − 0.951·9-s + 0.904·11-s − 1.34·13-s + 1.02·17-s + 0.830·19-s + 0.250·21-s + 0.257·23-s − 0.430·27-s + 1.24·29-s − 0.914·31-s + 0.199·33-s + 0.609·37-s − 0.296·39-s − 0.468·41-s + 1.37·43-s + 1.21·47-s + 0.285·49-s + 0.226·51-s + 0.634·53-s + 0.183·57-s − 0.539·59-s − 0.779·61-s − 1.07·63-s + 1.69·67-s + 0.0568·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.127054950\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.127054950\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 0.381T + 3T^{2} \) |
| 7 | \( 1 - 3T + 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 + 4.85T + 13T^{2} \) |
| 17 | \( 1 - 4.23T + 17T^{2} \) |
| 19 | \( 1 - 3.61T + 19T^{2} \) |
| 23 | \( 1 - 1.23T + 23T^{2} \) |
| 29 | \( 1 - 6.70T + 29T^{2} \) |
| 31 | \( 1 + 5.09T + 31T^{2} \) |
| 37 | \( 1 - 3.70T + 37T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 - 9T + 43T^{2} \) |
| 47 | \( 1 - 8.32T + 47T^{2} \) |
| 53 | \( 1 - 4.61T + 53T^{2} \) |
| 59 | \( 1 + 4.14T + 59T^{2} \) |
| 61 | \( 1 + 6.09T + 61T^{2} \) |
| 67 | \( 1 - 13.8T + 67T^{2} \) |
| 71 | \( 1 - 3T + 71T^{2} \) |
| 73 | \( 1 - 1.85T + 73T^{2} \) |
| 79 | \( 1 + 0.527T + 79T^{2} \) |
| 83 | \( 1 + 0.472T + 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 - 7.85T + 97T^{2} \) |
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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
−9.172890960323979301485505632484, −8.315174248138029414188008165210, −7.68357148473548296524574294449, −6.99028357239620489701010918243, −5.79004408028456986157731540758, −5.18327712807557257280538544047, −4.31189921040541195061219206601, −3.19699374833604956945701032047, −2.26807034994711060226495443523, −1.01082712045517129005682055579,
1.01082712045517129005682055579, 2.26807034994711060226495443523, 3.19699374833604956945701032047, 4.31189921040541195061219206601, 5.18327712807557257280538544047, 5.79004408028456986157731540758, 6.99028357239620489701010918243, 7.68357148473548296524574294449, 8.315174248138029414188008165210, 9.172890960323979301485505632484
