| L(s) = 1 | + 37.6·3-s + 5.31e3·7-s − 1.82e4·9-s − 1.04e4·11-s − 7.96e4·13-s + 3.13e5·17-s + 2.46e5·19-s + 2.00e5·21-s − 7.21e5·23-s − 1.42e6·27-s + 2.56e6·29-s + 3.29e6·31-s − 3.92e5·33-s + 1.40e7·37-s − 2.99e6·39-s + 1.70e7·41-s − 2.92e7·43-s − 4.10e7·47-s − 1.21e7·49-s + 1.18e7·51-s − 5.67e7·53-s + 9.29e6·57-s − 1.60e8·59-s + 5.33e7·61-s − 9.70e7·63-s + 2.80e8·67-s − 2.71e7·69-s + ⋯ |
| L(s) = 1 | + 0.268·3-s + 0.836·7-s − 0.928·9-s − 0.214·11-s − 0.773·13-s + 0.911·17-s + 0.434·19-s + 0.224·21-s − 0.537·23-s − 0.517·27-s + 0.674·29-s + 0.640·31-s − 0.0576·33-s + 1.23·37-s − 0.207·39-s + 0.941·41-s − 1.30·43-s − 1.22·47-s − 0.299·49-s + 0.244·51-s − 0.987·53-s + 0.116·57-s − 1.72·59-s + 0.492·61-s − 0.776·63-s + 1.70·67-s − 0.144·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 - 37.6T + 1.96e4T^{2} \) |
| 7 | \( 1 - 5.31e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 1.04e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 7.96e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 3.13e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 2.46e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 7.21e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 2.56e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 3.29e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.40e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.70e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.92e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 4.10e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 5.67e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.60e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 5.33e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.80e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 8.97e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 7.60e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + 4.10e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 5.21e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 2.37e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 6.03e8T + 7.60e17T^{2} \) |
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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
−9.344723215067258111081131130578, −8.064751532229781817994315566027, −7.894582807243512636171569866897, −6.46001592587069384289387422478, −5.40267902038517076470457995477, −4.62714638246660688034264230693, −3.27186830503529233962603443077, −2.40372286302610623437790082245, −1.23147645084858604465963368524, 0,
1.23147645084858604465963368524, 2.40372286302610623437790082245, 3.27186830503529233962603443077, 4.62714638246660688034264230693, 5.40267902038517076470457995477, 6.46001592587069384289387422478, 7.894582807243512636171569866897, 8.064751532229781817994315566027, 9.344723215067258111081131130578
