L(s) = 1 | − 2.56·2-s + 1.12·3-s + 4.57·4-s − 1.24·5-s − 2.88·6-s − 2.50·7-s − 6.60·8-s − 1.73·9-s + 3.20·10-s − 11-s + 5.15·12-s + 6.42·14-s − 1.40·15-s + 7.78·16-s + 2.78·17-s + 4.43·18-s + 5.79·19-s − 5.70·20-s − 2.82·21-s + 2.56·22-s − 8.60·23-s − 7.44·24-s − 3.44·25-s − 5.33·27-s − 11.4·28-s − 0.584·29-s + 3.60·30-s + ⋯ |
L(s) = 1 | − 1.81·2-s + 0.650·3-s + 2.28·4-s − 0.558·5-s − 1.17·6-s − 0.946·7-s − 2.33·8-s − 0.576·9-s + 1.01·10-s − 0.301·11-s + 1.48·12-s + 1.71·14-s − 0.363·15-s + 1.94·16-s + 0.676·17-s + 1.04·18-s + 1.32·19-s − 1.27·20-s − 0.615·21-s + 0.546·22-s − 1.79·23-s − 1.51·24-s − 0.688·25-s − 1.02·27-s − 2.16·28-s − 0.108·29-s + 0.658·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5025341108\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5025341108\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 2.56T + 2T^{2} \) |
| 3 | \( 1 - 1.12T + 3T^{2} \) |
| 5 | \( 1 + 1.24T + 5T^{2} \) |
| 7 | \( 1 + 2.50T + 7T^{2} \) |
| 17 | \( 1 - 2.78T + 17T^{2} \) |
| 19 | \( 1 - 5.79T + 19T^{2} \) |
| 23 | \( 1 + 8.60T + 23T^{2} \) |
| 29 | \( 1 + 0.584T + 29T^{2} \) |
| 31 | \( 1 - 5.56T + 31T^{2} \) |
| 37 | \( 1 + 8.21T + 37T^{2} \) |
| 41 | \( 1 + 0.167T + 41T^{2} \) |
| 43 | \( 1 - 9.05T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 + 1.39T + 53T^{2} \) |
| 59 | \( 1 + 5.61T + 59T^{2} \) |
| 61 | \( 1 - 12.3T + 61T^{2} \) |
| 67 | \( 1 - 10.9T + 67T^{2} \) |
| 71 | \( 1 + 3.69T + 71T^{2} \) |
| 73 | \( 1 + 7.96T + 73T^{2} \) |
| 79 | \( 1 - 14.8T + 79T^{2} \) |
| 83 | \( 1 + 9.73T + 83T^{2} \) |
| 89 | \( 1 + 0.577T + 89T^{2} \) |
| 97 | \( 1 - 3.91T + 97T^{2} \) |
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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
−9.173754497449977080360382136691, −8.520843825615340589261308400450, −7.73742144059281387228110388440, −7.47968108609013068207837737367, −6.33789923832658276743409574761, −5.60346070554531244205400829818, −3.78082263785277393306178081769, −3.01166328285175837819117026931, −2.09865294457301223725348875668, −0.57941625691200101281745925073,
0.57941625691200101281745925073, 2.09865294457301223725348875668, 3.01166328285175837819117026931, 3.78082263785277393306178081769, 5.60346070554531244205400829818, 6.33789923832658276743409574761, 7.47968108609013068207837737367, 7.73742144059281387228110388440, 8.520843825615340589261308400450, 9.173754497449977080360382136691