L(s) = 1 | − 2.81·2-s − 0.519·3-s + 5.90·4-s − 3.87·5-s + 1.45·6-s + 3.50·7-s − 10.9·8-s − 2.73·9-s + 10.8·10-s − 11-s − 3.06·12-s + 0.0114·13-s − 9.86·14-s + 2.01·15-s + 19.0·16-s + 17-s + 7.67·18-s + 1.70·19-s − 22.8·20-s − 1.82·21-s + 2.81·22-s + 2.63·23-s + 5.69·24-s + 10.0·25-s − 0.0320·26-s + 2.97·27-s + 20.7·28-s + ⋯ |
L(s) = 1 | − 1.98·2-s − 0.299·3-s + 2.95·4-s − 1.73·5-s + 0.596·6-s + 1.32·7-s − 3.87·8-s − 0.910·9-s + 3.44·10-s − 0.301·11-s − 0.885·12-s + 0.00316·13-s − 2.63·14-s + 0.519·15-s + 4.76·16-s + 0.242·17-s + 1.80·18-s + 0.391·19-s − 5.11·20-s − 0.397·21-s + 0.599·22-s + 0.549·23-s + 1.16·24-s + 2.00·25-s − 0.00629·26-s + 0.572·27-s + 3.91·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 43 | \( 1 - T \) |
good | 2 | \( 1 + 2.81T + 2T^{2} \) |
| 3 | \( 1 + 0.519T + 3T^{2} \) |
| 5 | \( 1 + 3.87T + 5T^{2} \) |
| 7 | \( 1 - 3.50T + 7T^{2} \) |
| 13 | \( 1 - 0.0114T + 13T^{2} \) |
| 19 | \( 1 - 1.70T + 19T^{2} \) |
| 23 | \( 1 - 2.63T + 23T^{2} \) |
| 29 | \( 1 - 4.94T + 29T^{2} \) |
| 31 | \( 1 + 6.86T + 31T^{2} \) |
| 37 | \( 1 + 6.53T + 37T^{2} \) |
| 41 | \( 1 + 1.91T + 41T^{2} \) |
| 47 | \( 1 + 7.40T + 47T^{2} \) |
| 53 | \( 1 - 0.430T + 53T^{2} \) |
| 59 | \( 1 + 12.5T + 59T^{2} \) |
| 61 | \( 1 - 5.35T + 61T^{2} \) |
| 67 | \( 1 - 2.50T + 67T^{2} \) |
| 71 | \( 1 + 1.55T + 71T^{2} \) |
| 73 | \( 1 - 4.21T + 73T^{2} \) |
| 79 | \( 1 + 6.39T + 79T^{2} \) |
| 83 | \( 1 - 11.0T + 83T^{2} \) |
| 89 | \( 1 - 1.27T + 89T^{2} \) |
| 97 | \( 1 + 10.2T + 97T^{2} \) |
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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
−7.77315130263066014992204232103, −7.21674616838443958123859407085, −6.53577023675279829655239384658, −5.49834437843037074386225273576, −4.83536618962838748533104566964, −3.52346984167947592312351481766, −2.93471595201485292757198235152, −1.82849279318070515402602225627, −0.865448243951044420225714632782, 0,
0.865448243951044420225714632782, 1.82849279318070515402602225627, 2.93471595201485292757198235152, 3.52346984167947592312351481766, 4.83536618962838748533104566964, 5.49834437843037074386225273576, 6.53577023675279829655239384658, 7.21674616838443958123859407085, 7.77315130263066014992204232103