| L(s) = 1 | − 2-s + 0.441·3-s + 4-s + 3.09·5-s − 0.441·6-s − 4.79·7-s − 8-s − 2.80·9-s − 3.09·10-s + 5.99·11-s + 0.441·12-s + 1.20·13-s + 4.79·14-s + 1.36·15-s + 16-s − 1.57·17-s + 2.80·18-s − 1.11·19-s + 3.09·20-s − 2.11·21-s − 5.99·22-s − 2.43·23-s − 0.441·24-s + 4.56·25-s − 1.20·26-s − 2.56·27-s − 4.79·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.254·3-s + 0.5·4-s + 1.38·5-s − 0.180·6-s − 1.81·7-s − 0.353·8-s − 0.935·9-s − 0.978·10-s + 1.80·11-s + 0.127·12-s + 0.334·13-s + 1.28·14-s + 0.352·15-s + 0.250·16-s − 0.381·17-s + 0.661·18-s − 0.255·19-s + 0.691·20-s − 0.461·21-s − 1.27·22-s − 0.507·23-s − 0.0900·24-s + 0.913·25-s − 0.236·26-s − 0.492·27-s − 0.906·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 2011 | \( 1 + T \) |
| good | 3 | \( 1 - 0.441T + 3T^{2} \) |
| 5 | \( 1 - 3.09T + 5T^{2} \) |
| 7 | \( 1 + 4.79T + 7T^{2} \) |
| 11 | \( 1 - 5.99T + 11T^{2} \) |
| 13 | \( 1 - 1.20T + 13T^{2} \) |
| 17 | \( 1 + 1.57T + 17T^{2} \) |
| 19 | \( 1 + 1.11T + 19T^{2} \) |
| 23 | \( 1 + 2.43T + 23T^{2} \) |
| 29 | \( 1 + 1.42T + 29T^{2} \) |
| 31 | \( 1 + 6.94T + 31T^{2} \) |
| 37 | \( 1 + 7.52T + 37T^{2} \) |
| 41 | \( 1 - 3.83T + 41T^{2} \) |
| 43 | \( 1 - 4.99T + 43T^{2} \) |
| 47 | \( 1 + 6.63T + 47T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 - 2.03T + 59T^{2} \) |
| 61 | \( 1 + 0.846T + 61T^{2} \) |
| 67 | \( 1 - 1.05T + 67T^{2} \) |
| 71 | \( 1 - 9.23T + 71T^{2} \) |
| 73 | \( 1 + 9.98T + 73T^{2} \) |
| 79 | \( 1 - 13.1T + 79T^{2} \) |
| 83 | \( 1 + 8.86T + 83T^{2} \) |
| 89 | \( 1 + 17.6T + 89T^{2} \) |
| 97 | \( 1 + 3.35T + 97T^{2} \) |
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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
−8.423881977904677583597483180797, −7.14752593636017242161341801776, −6.48076138725494249535326690621, −6.15122645428139745661984312908, −5.51541408375015494125683164543, −3.91931143008017412929567993025, −3.26827124856301806543588527599, −2.35037224268392869072109338081, −1.47278267985871141845117522359, 0,
1.47278267985871141845117522359, 2.35037224268392869072109338081, 3.26827124856301806543588527599, 3.91931143008017412929567993025, 5.51541408375015494125683164543, 6.15122645428139745661984312908, 6.48076138725494249535326690621, 7.14752593636017242161341801776, 8.423881977904677583597483180797
