| L(s) = 1 | − 0.881·2-s − 3-s − 1.22·4-s + 2.92·5-s + 0.881·6-s + 2.84·8-s + 9-s − 2.58·10-s + 3.24·11-s + 1.22·12-s − 1.62·13-s − 2.92·15-s − 0.0586·16-s − 4.45·17-s − 0.881·18-s + 1.70·19-s − 3.57·20-s − 2.86·22-s − 23-s − 2.84·24-s + 3.56·25-s + 1.43·26-s − 27-s − 8.39·29-s + 2.58·30-s − 3.59·31-s − 5.63·32-s + ⋯ |
| L(s) = 1 | − 0.623·2-s − 0.577·3-s − 0.611·4-s + 1.30·5-s + 0.359·6-s + 1.00·8-s + 0.333·9-s − 0.815·10-s + 0.979·11-s + 0.353·12-s − 0.451·13-s − 0.755·15-s − 0.0146·16-s − 1.08·17-s − 0.207·18-s + 0.390·19-s − 0.800·20-s − 0.610·22-s − 0.208·23-s − 0.579·24-s + 0.713·25-s + 0.281·26-s − 0.192·27-s − 1.55·29-s + 0.471·30-s − 0.646·31-s − 0.995·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + 0.881T + 2T^{2} \) |
| 5 | \( 1 - 2.92T + 5T^{2} \) |
| 11 | \( 1 - 3.24T + 11T^{2} \) |
| 13 | \( 1 + 1.62T + 13T^{2} \) |
| 17 | \( 1 + 4.45T + 17T^{2} \) |
| 19 | \( 1 - 1.70T + 19T^{2} \) |
| 29 | \( 1 + 8.39T + 29T^{2} \) |
| 31 | \( 1 + 3.59T + 31T^{2} \) |
| 37 | \( 1 + 1.06T + 37T^{2} \) |
| 41 | \( 1 + 6.59T + 41T^{2} \) |
| 43 | \( 1 + 6.47T + 43T^{2} \) |
| 47 | \( 1 - 5.36T + 47T^{2} \) |
| 53 | \( 1 + 0.262T + 53T^{2} \) |
| 59 | \( 1 + 4.00T + 59T^{2} \) |
| 61 | \( 1 + 3.00T + 61T^{2} \) |
| 67 | \( 1 - 3.32T + 67T^{2} \) |
| 71 | \( 1 - 1.66T + 71T^{2} \) |
| 73 | \( 1 + 0.422T + 73T^{2} \) |
| 79 | \( 1 - 0.231T + 79T^{2} \) |
| 83 | \( 1 + 17.3T + 83T^{2} \) |
| 89 | \( 1 + 0.971T + 89T^{2} \) |
| 97 | \( 1 - 19.0T + 97T^{2} \) |
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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
−8.509442796284707908079840234342, −7.40999882020296345738670607070, −6.79064904114552639437320187420, −5.93655463268485446630075358111, −5.29240801513147903448618427334, −4.52260995611397252407910354464, −3.61737579637124197463270127585, −2.07681963749251648879172632082, −1.41919264702191898624766132447, 0,
1.41919264702191898624766132447, 2.07681963749251648879172632082, 3.61737579637124197463270127585, 4.52260995611397252407910354464, 5.29240801513147903448618427334, 5.93655463268485446630075358111, 6.79064904114552639437320187420, 7.40999882020296345738670607070, 8.509442796284707908079840234342
