| L(s) = 1 | − 10.6·3-s − 128.·7-s − 129.·9-s + 117.·11-s + 389.·13-s − 910.·17-s − 675.·19-s + 1.37e3·21-s + 3.66e3·23-s + 3.97e3·27-s + 2.85e3·29-s + 240.·31-s − 1.25e3·33-s − 430.·37-s − 4.15e3·39-s − 6.96e3·41-s + 6.74e3·43-s + 4.61e3·47-s − 262.·49-s + 9.72e3·51-s + 1.24e4·53-s + 7.21e3·57-s + 2.17e4·59-s − 2.82e3·61-s + 1.66e4·63-s + 5.67e4·67-s − 3.91e4·69-s + ⋯ |
| L(s) = 1 | − 0.684·3-s − 0.992·7-s − 0.531·9-s + 0.293·11-s + 0.639·13-s − 0.764·17-s − 0.429·19-s + 0.679·21-s + 1.44·23-s + 1.04·27-s + 0.630·29-s + 0.0449·31-s − 0.201·33-s − 0.0517·37-s − 0.437·39-s − 0.647·41-s + 0.556·43-s + 0.304·47-s − 0.0156·49-s + 0.523·51-s + 0.610·53-s + 0.293·57-s + 0.814·59-s − 0.0972·61-s + 0.527·63-s + 1.54·67-s − 0.988·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 10.6T + 243T^{2} \) |
| 7 | \( 1 + 128.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 117.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 389.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 910.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 675.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.66e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.85e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 240.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 430.T + 6.93e7T^{2} \) |
| 41 | \( 1 + 6.96e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 6.74e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 4.61e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.24e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.17e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.82e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.67e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.90e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.59e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.33e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 8.49e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.00e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.30e4T + 8.58e9T^{2} \) |
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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
−9.006565544608003598306617571544, −8.477931162669325580277492542612, −7.02611028220229371224241237364, −6.46032098533090432255487850595, −5.69097856743631595393247947628, −4.66770944560939017937944944897, −3.52605929049086609499102635756, −2.56584739804725677809785339191, −1.00518492350298908666734206287, 0,
1.00518492350298908666734206287, 2.56584739804725677809785339191, 3.52605929049086609499102635756, 4.66770944560939017937944944897, 5.69097856743631595393247947628, 6.46032098533090432255487850595, 7.02611028220229371224241237364, 8.477931162669325580277492542612, 9.006565544608003598306617571544
