| L(s) = 1 | − 0.743·2-s + 3-s − 1.44·4-s − 3.36·5-s − 0.743·6-s − 2.98·7-s + 2.56·8-s + 9-s + 2.49·10-s + 1.73·11-s − 1.44·12-s + 0.418·13-s + 2.21·14-s − 3.36·15-s + 0.989·16-s + 3.04·17-s − 0.743·18-s + 6.79·19-s + 4.86·20-s − 2.98·21-s − 1.28·22-s + 23-s + 2.56·24-s + 6.30·25-s − 0.311·26-s + 27-s + 4.31·28-s + ⋯ |
| L(s) = 1 | − 0.525·2-s + 0.577·3-s − 0.723·4-s − 1.50·5-s − 0.303·6-s − 1.12·7-s + 0.906·8-s + 0.333·9-s + 0.790·10-s + 0.522·11-s − 0.417·12-s + 0.116·13-s + 0.592·14-s − 0.868·15-s + 0.247·16-s + 0.737·17-s − 0.175·18-s + 1.55·19-s + 1.08·20-s − 0.651·21-s − 0.274·22-s + 0.208·23-s + 0.523·24-s + 1.26·25-s − 0.0610·26-s + 0.192·27-s + 0.816·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 0.743T + 2T^{2} \) |
| 5 | \( 1 + 3.36T + 5T^{2} \) |
| 7 | \( 1 + 2.98T + 7T^{2} \) |
| 11 | \( 1 - 1.73T + 11T^{2} \) |
| 13 | \( 1 - 0.418T + 13T^{2} \) |
| 17 | \( 1 - 3.04T + 17T^{2} \) |
| 19 | \( 1 - 6.79T + 19T^{2} \) |
| 31 | \( 1 + 1.26T + 31T^{2} \) |
| 37 | \( 1 + 8.40T + 37T^{2} \) |
| 41 | \( 1 + 5.67T + 41T^{2} \) |
| 43 | \( 1 - 2.38T + 43T^{2} \) |
| 47 | \( 1 + 1.01T + 47T^{2} \) |
| 53 | \( 1 - 1.08T + 53T^{2} \) |
| 59 | \( 1 + 0.434T + 59T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 + 4.29T + 67T^{2} \) |
| 71 | \( 1 + 7.01T + 71T^{2} \) |
| 73 | \( 1 + 9.28T + 73T^{2} \) |
| 79 | \( 1 - 2.43T + 79T^{2} \) |
| 83 | \( 1 - 15.3T + 83T^{2} \) |
| 89 | \( 1 - 11.8T + 89T^{2} \) |
| 97 | \( 1 - 4.37T + 97T^{2} \) |
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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
−8.886894866680530317670249195322, −7.964407955418548135731818354437, −7.50185496541243731499986219207, −6.76735393239233091629900919114, −5.44965017917238054918728589553, −4.45387690681618710874905565352, −3.50023417849304708803727768226, −3.27851684412075141378758473367, −1.24522085264184878085266314942, 0,
1.24522085264184878085266314942, 3.27851684412075141378758473367, 3.50023417849304708803727768226, 4.45387690681618710874905565352, 5.44965017917238054918728589553, 6.76735393239233091629900919114, 7.50185496541243731499986219207, 7.964407955418548135731818354437, 8.886894866680530317670249195322
