| 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
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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
−9.730380900495822673922702747116, −8.924812847793571753205342717778, −8.524308617634012615105037056956, −7.07757333880905550400463489860, −6.62240857839917143686722720799, −4.86157997887748385992612553422, −3.75411952846061902092400520960, −2.82259954042516685527710507584, −1.85703357461536510586393077429, 0,
1.85703357461536510586393077429, 2.82259954042516685527710507584, 3.75411952846061902092400520960, 4.86157997887748385992612553422, 6.62240857839917143686722720799, 7.07757333880905550400463489860, 8.524308617634012615105037056956, 8.924812847793571753205342717778, 9.730380900495822673922702747116
