| L(s) = 1 | + 0.656·2-s − 1.56·4-s − 7-s − 2.34·8-s + 1.91·11-s − 0.255·13-s − 0.656·14-s + 1.59·16-s − 6.16·17-s + 7.08·19-s + 1.25·22-s + 2.56·23-s − 0.167·26-s + 1.56·28-s − 2.56·29-s − 1.05·31-s + 5.73·32-s − 4.04·34-s − 8.39·37-s + 4.64·38-s + 10.8·41-s + 5.08·43-s − 3·44-s + 1.68·46-s − 3.05·47-s + 49-s + 0.401·52-s + ⋯ |
| L(s) = 1 | + 0.464·2-s − 0.784·4-s − 0.377·7-s − 0.828·8-s + 0.576·11-s − 0.0708·13-s − 0.175·14-s + 0.399·16-s − 1.49·17-s + 1.62·19-s + 0.267·22-s + 0.535·23-s − 0.0329·26-s + 0.296·28-s − 0.477·29-s − 0.189·31-s + 1.01·32-s − 0.694·34-s − 1.37·37-s + 0.754·38-s + 1.69·41-s + 0.775·43-s − 0.452·44-s + 0.248·46-s − 0.446·47-s + 0.142·49-s + 0.0556·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 2 | \( 1 - 0.656T + 2T^{2} \) |
| 11 | \( 1 - 1.91T + 11T^{2} \) |
| 13 | \( 1 + 0.255T + 13T^{2} \) |
| 17 | \( 1 + 6.16T + 17T^{2} \) |
| 19 | \( 1 - 7.08T + 19T^{2} \) |
| 23 | \( 1 - 2.56T + 23T^{2} \) |
| 29 | \( 1 + 2.56T + 29T^{2} \) |
| 31 | \( 1 + 1.05T + 31T^{2} \) |
| 37 | \( 1 + 8.39T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 - 5.08T + 43T^{2} \) |
| 47 | \( 1 + 3.05T + 47T^{2} \) |
| 53 | \( 1 + 2.77T + 53T^{2} \) |
| 59 | \( 1 + 8.36T + 59T^{2} \) |
| 61 | \( 1 + 12.3T + 61T^{2} \) |
| 67 | \( 1 + 8.93T + 67T^{2} \) |
| 71 | \( 1 - 12.5T + 71T^{2} \) |
| 73 | \( 1 - 9.96T + 73T^{2} \) |
| 79 | \( 1 + 9.96T + 79T^{2} \) |
| 83 | \( 1 + 3.71T + 83T^{2} \) |
| 89 | \( 1 + 6.33T + 89T^{2} \) |
| 97 | \( 1 - 5.64T + 97T^{2} \) |
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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
−7.904866259964254889485124035790, −7.15221649533612235153521580192, −6.38131682750270174057663849800, −5.64864079580846044290746332326, −4.91326204875044506268445858442, −4.20316801294611183965617604339, −3.46462548405289852447864186265, −2.67033165243441128499796355716, −1.29408087486877157715752612739, 0,
1.29408087486877157715752612739, 2.67033165243441128499796355716, 3.46462548405289852447864186265, 4.20316801294611183965617604339, 4.91326204875044506268445858442, 5.64864079580846044290746332326, 6.38131682750270174057663849800, 7.15221649533612235153521580192, 7.904866259964254889485124035790
