| L(s) = 1 | + 2.23·5-s − 3.70·11-s + 6.86·13-s − 7.47·17-s − 5.45·19-s − 3.16·23-s + 3.70·29-s + 6.86·31-s − 8.70·37-s + 2.23·41-s − 4.70·43-s − 13.4·47-s − 5.32·53-s − 8.27·55-s + 5.94·59-s + 0.206·61-s + 15.3·65-s − 9.70·67-s − 5.24·71-s − 1.41·73-s + 5·79-s − 8.23·83-s − 16.7·85-s − 11.2·89-s − 12.1·95-s − 0.206·97-s + 3.70·101-s + ⋯ |
| L(s) = 1 | + 0.999·5-s − 1.11·11-s + 1.90·13-s − 1.81·17-s − 1.25·19-s − 0.659·23-s + 0.687·29-s + 1.23·31-s − 1.43·37-s + 0.349·41-s − 0.717·43-s − 1.96·47-s − 0.731·53-s − 1.11·55-s + 0.773·59-s + 0.0264·61-s + 1.90·65-s − 1.18·67-s − 0.622·71-s − 0.165·73-s + 0.562·79-s − 0.904·83-s − 1.81·85-s − 1.19·89-s − 1.25·95-s − 0.0209·97-s + 0.368·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 2.23T + 5T^{2} \) |
| 11 | \( 1 + 3.70T + 11T^{2} \) |
| 13 | \( 1 - 6.86T + 13T^{2} \) |
| 17 | \( 1 + 7.47T + 17T^{2} \) |
| 19 | \( 1 + 5.45T + 19T^{2} \) |
| 23 | \( 1 + 3.16T + 23T^{2} \) |
| 29 | \( 1 - 3.70T + 29T^{2} \) |
| 31 | \( 1 - 6.86T + 31T^{2} \) |
| 37 | \( 1 + 8.70T + 37T^{2} \) |
| 41 | \( 1 - 2.23T + 41T^{2} \) |
| 43 | \( 1 + 4.70T + 43T^{2} \) |
| 47 | \( 1 + 13.4T + 47T^{2} \) |
| 53 | \( 1 + 5.32T + 53T^{2} \) |
| 59 | \( 1 - 5.94T + 59T^{2} \) |
| 61 | \( 1 - 0.206T + 61T^{2} \) |
| 67 | \( 1 + 9.70T + 67T^{2} \) |
| 71 | \( 1 + 5.24T + 71T^{2} \) |
| 73 | \( 1 + 1.41T + 73T^{2} \) |
| 79 | \( 1 - 5T + 79T^{2} \) |
| 83 | \( 1 + 8.23T + 83T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 + 0.206T + 97T^{2} \) |
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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
−8.171182587164797127916946521061, −6.84244516639305900980333480575, −6.32726151769885552381878622280, −5.87599586192546648664441282065, −4.88427059100475437462964298917, −4.22678803347179990450392850176, −3.17884702050383642090873199213, −2.23917184697956027520804617929, −1.57870253885654168756377608683, 0,
1.57870253885654168756377608683, 2.23917184697956027520804617929, 3.17884702050383642090873199213, 4.22678803347179990450392850176, 4.88427059100475437462964298917, 5.87599586192546648664441282065, 6.32726151769885552381878622280, 6.84244516639305900980333480575, 8.171182587164797127916946521061
