| L(s) = 1 | − 22.6·3-s + 107.·7-s + 269.·9-s + 290.·11-s + 519.·13-s + 2.03e3·17-s − 1.70e3·19-s − 2.42e3·21-s − 4.35e3·23-s − 588.·27-s + 4.79e3·29-s − 7.59e3·31-s − 6.58e3·33-s + 5.26e3·37-s − 1.17e4·39-s − 1.01e4·41-s + 2.34e4·43-s + 3.31e3·47-s − 5.30e3·49-s − 4.59e4·51-s − 2.89e4·53-s + 3.86e4·57-s − 1.45e4·59-s + 2.43e4·61-s + 2.88e4·63-s + 5.93e3·67-s + 9.85e4·69-s + ⋯ |
| L(s) = 1 | − 1.45·3-s + 0.827·7-s + 1.10·9-s + 0.724·11-s + 0.852·13-s + 1.70·17-s − 1.08·19-s − 1.20·21-s − 1.71·23-s − 0.155·27-s + 1.05·29-s − 1.42·31-s − 1.05·33-s + 0.631·37-s − 1.23·39-s − 0.939·41-s + 1.93·43-s + 0.219·47-s − 0.315·49-s − 2.47·51-s − 1.41·53-s + 1.57·57-s − 0.544·59-s + 0.838·61-s + 0.915·63-s + 0.161·67-s + 2.49·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.425437206\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.425437206\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 22.6T + 243T^{2} \) |
| 7 | \( 1 - 107.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 290.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 519.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.03e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.70e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.35e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.79e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.59e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.26e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.01e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.34e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 3.31e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.89e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.45e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.43e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.93e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.44e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.81e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.16e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.73e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 9.05e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 7.38e4T + 8.58e9T^{2} \) |
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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
−10.70546166754514926057359456111, −9.822621132596554465852484926147, −8.523567997443469871369269859809, −7.61797383286529513090051346964, −6.31522371005774464810065066987, −5.82194519711181739413188926583, −4.74496576163477167868638072845, −3.74151781030574801539881639882, −1.73341643697887484934076044306, −0.71843029474451144503081885243,
0.71843029474451144503081885243, 1.73341643697887484934076044306, 3.74151781030574801539881639882, 4.74496576163477167868638072845, 5.82194519711181739413188926583, 6.31522371005774464810065066987, 7.61797383286529513090051346964, 8.523567997443469871369269859809, 9.822621132596554465852484926147, 10.70546166754514926057359456111
