| L(s) = 1 | + 0.184·2-s − 1.96·4-s + 5-s + 2.68·7-s − 0.732·8-s + 0.184·10-s − 4.77·11-s + 6.17·13-s + 0.496·14-s + 3.79·16-s + 5.76·17-s + 7.05·19-s − 1.96·20-s − 0.881·22-s − 0.693·23-s + 25-s + 1.14·26-s − 5.28·28-s − 8.42·29-s + 3.38·31-s + 2.16·32-s + 1.06·34-s + 2.68·35-s − 11.0·37-s + 1.30·38-s − 0.732·40-s − 2.52·41-s + ⋯ |
| L(s) = 1 | + 0.130·2-s − 0.982·4-s + 0.447·5-s + 1.01·7-s − 0.258·8-s + 0.0583·10-s − 1.43·11-s + 1.71·13-s + 0.132·14-s + 0.949·16-s + 1.39·17-s + 1.61·19-s − 0.439·20-s − 0.187·22-s − 0.144·23-s + 0.200·25-s + 0.223·26-s − 0.998·28-s − 1.56·29-s + 0.608·31-s + 0.382·32-s + 0.182·34-s + 0.454·35-s − 1.82·37-s + 0.211·38-s − 0.115·40-s − 0.394·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.103836876\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.103836876\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 89 | \( 1 + T \) |
| good | 2 | \( 1 - 0.184T + 2T^{2} \) |
| 7 | \( 1 - 2.68T + 7T^{2} \) |
| 11 | \( 1 + 4.77T + 11T^{2} \) |
| 13 | \( 1 - 6.17T + 13T^{2} \) |
| 17 | \( 1 - 5.76T + 17T^{2} \) |
| 19 | \( 1 - 7.05T + 19T^{2} \) |
| 23 | \( 1 + 0.693T + 23T^{2} \) |
| 29 | \( 1 + 8.42T + 29T^{2} \) |
| 31 | \( 1 - 3.38T + 31T^{2} \) |
| 37 | \( 1 + 11.0T + 37T^{2} \) |
| 41 | \( 1 + 2.52T + 41T^{2} \) |
| 43 | \( 1 + 4.09T + 43T^{2} \) |
| 47 | \( 1 - 0.000463T + 47T^{2} \) |
| 53 | \( 1 - 4.86T + 53T^{2} \) |
| 59 | \( 1 - 0.626T + 59T^{2} \) |
| 61 | \( 1 - 9.47T + 61T^{2} \) |
| 67 | \( 1 + 0.700T + 67T^{2} \) |
| 71 | \( 1 - 9.55T + 71T^{2} \) |
| 73 | \( 1 - 9.93T + 73T^{2} \) |
| 79 | \( 1 - 3.17T + 79T^{2} \) |
| 83 | \( 1 + 5.19T + 83T^{2} \) |
| 97 | \( 1 - 4.50T + 97T^{2} \) |
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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
−8.226153282635924797514635874331, −8.057405418923069177462068470292, −7.14033219094035972073408949763, −5.81944293455015328765784606470, −5.36811856694501022575981131556, −4.99355958831961243059379311033, −3.71524795216309296035306555634, −3.23957201139011179999355363094, −1.79120295369070775164245103192, −0.879624285376175408973533641841,
0.879624285376175408973533641841, 1.79120295369070775164245103192, 3.23957201139011179999355363094, 3.71524795216309296035306555634, 4.99355958831961243059379311033, 5.36811856694501022575981131556, 5.81944293455015328765784606470, 7.14033219094035972073408949763, 8.057405418923069177462068470292, 8.226153282635924797514635874331
