L(s) = 1 | − 2-s + 2.05·3-s + 4-s + 3.66·5-s − 2.05·6-s + 4.21·7-s − 8-s + 1.23·9-s − 3.66·10-s − 3.07·11-s + 2.05·12-s + 3.04·13-s − 4.21·14-s + 7.53·15-s + 16-s − 6.40·17-s − 1.23·18-s + 3.39·19-s + 3.66·20-s + 8.67·21-s + 3.07·22-s − 23-s − 2.05·24-s + 8.40·25-s − 3.04·26-s − 3.63·27-s + 4.21·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.18·3-s + 0.5·4-s + 1.63·5-s − 0.840·6-s + 1.59·7-s − 0.353·8-s + 0.411·9-s − 1.15·10-s − 0.925·11-s + 0.593·12-s + 0.845·13-s − 1.12·14-s + 1.94·15-s + 0.250·16-s − 1.55·17-s − 0.290·18-s + 0.779·19-s + 0.818·20-s + 1.89·21-s + 0.654·22-s − 0.208·23-s − 0.420·24-s + 1.68·25-s − 0.597·26-s − 0.699·27-s + 0.796·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.692765011\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.692765011\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
good | 3 | \( 1 - 2.05T + 3T^{2} \) |
| 5 | \( 1 - 3.66T + 5T^{2} \) |
| 7 | \( 1 - 4.21T + 7T^{2} \) |
| 11 | \( 1 + 3.07T + 11T^{2} \) |
| 13 | \( 1 - 3.04T + 13T^{2} \) |
| 17 | \( 1 + 6.40T + 17T^{2} \) |
| 19 | \( 1 - 3.39T + 19T^{2} \) |
| 31 | \( 1 + 1.07T + 31T^{2} \) |
| 37 | \( 1 + 0.533T + 37T^{2} \) |
| 41 | \( 1 + 2.53T + 41T^{2} \) |
| 43 | \( 1 - 7.18T + 43T^{2} \) |
| 47 | \( 1 + 7.95T + 47T^{2} \) |
| 53 | \( 1 + 3.19T + 53T^{2} \) |
| 59 | \( 1 + 8.47T + 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 + 5.27T + 67T^{2} \) |
| 71 | \( 1 - 3.30T + 71T^{2} \) |
| 73 | \( 1 + 6.15T + 73T^{2} \) |
| 79 | \( 1 + 9.10T + 79T^{2} \) |
| 83 | \( 1 - 13.6T + 83T^{2} \) |
| 89 | \( 1 - 3.79T + 89T^{2} \) |
| 97 | \( 1 - 18.0T + 97T^{2} \) |
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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
−9.254102213276291831109507619816, −8.959635691093620089369674921780, −8.148693918844911453750084531201, −7.58231618108122537968963771479, −6.38058794786585620804954206741, −5.51496403373086113143247878558, −4.59625102915497028247356385856, −3.01137774704123708665246071223, −2.12401939565870333318809052285, −1.56709946558897551870475970795,
1.56709946558897551870475970795, 2.12401939565870333318809052285, 3.01137774704123708665246071223, 4.59625102915497028247356385856, 5.51496403373086113143247878558, 6.38058794786585620804954206741, 7.58231618108122537968963771479, 8.148693918844911453750084531201, 8.959635691093620089369674921780, 9.254102213276291831109507619816