| L(s) = 1 | + 2-s + 4-s + 0.732·5-s + 4.46·7-s + 8-s + 0.732·10-s + 2.26·11-s − 13-s + 4.46·14-s + 16-s + 6.92·17-s − 1.73·19-s + 0.732·20-s + 2.26·22-s − 6.19·23-s − 4.46·25-s − 26-s + 4.46·28-s + 6.26·29-s + 0.535·31-s + 32-s + 6.92·34-s + 3.26·35-s − 9.46·37-s − 1.73·38-s + 0.732·40-s + 4.73·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.327·5-s + 1.68·7-s + 0.353·8-s + 0.231·10-s + 0.683·11-s − 0.277·13-s + 1.19·14-s + 0.250·16-s + 1.68·17-s − 0.397·19-s + 0.163·20-s + 0.483·22-s − 1.29·23-s − 0.892·25-s − 0.196·26-s + 0.843·28-s + 1.16·29-s + 0.0962·31-s + 0.176·32-s + 1.18·34-s + 0.552·35-s − 1.55·37-s − 0.280·38-s + 0.115·40-s + 0.739·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2106 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2106 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.786948989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.786948989\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 0.732T + 5T^{2} \) |
| 7 | \( 1 - 4.46T + 7T^{2} \) |
| 11 | \( 1 - 2.26T + 11T^{2} \) |
| 17 | \( 1 - 6.92T + 17T^{2} \) |
| 19 | \( 1 + 1.73T + 19T^{2} \) |
| 23 | \( 1 + 6.19T + 23T^{2} \) |
| 29 | \( 1 - 6.26T + 29T^{2} \) |
| 31 | \( 1 - 0.535T + 31T^{2} \) |
| 37 | \( 1 + 9.46T + 37T^{2} \) |
| 41 | \( 1 - 4.73T + 41T^{2} \) |
| 43 | \( 1 + 8.92T + 43T^{2} \) |
| 47 | \( 1 - 7.46T + 47T^{2} \) |
| 53 | \( 1 + 8.66T + 53T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 + 7.19T + 61T^{2} \) |
| 67 | \( 1 - 8.39T + 67T^{2} \) |
| 71 | \( 1 - 3.53T + 71T^{2} \) |
| 73 | \( 1 + 6.53T + 73T^{2} \) |
| 79 | \( 1 - 16.1T + 79T^{2} \) |
| 83 | \( 1 + 2.66T + 83T^{2} \) |
| 89 | \( 1 + 3.80T + 89T^{2} \) |
| 97 | \( 1 - 12.7T + 97T^{2} \) |
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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
−9.045027683375592144851419701751, −7.996216773524625232654777294316, −7.76663063854727673989937412828, −6.58981646589436373780265739143, −5.77691896801710440851022057911, −5.06818337631653742009157536024, −4.33168323799140213052140138005, −3.42205865099647687718499154796, −2.10095238414441953504692234051, −1.36661264103879083457634835573,
1.36661264103879083457634835573, 2.10095238414441953504692234051, 3.42205865099647687718499154796, 4.33168323799140213052140138005, 5.06818337631653742009157536024, 5.77691896801710440851022057911, 6.58981646589436373780265739143, 7.76663063854727673989937412828, 7.996216773524625232654777294316, 9.045027683375592144851419701751
