| L(s) = 1 | − 1.46·2-s − 3-s + 0.139·4-s + 1.46·6-s + 7-s + 2.72·8-s + 9-s − 11-s − 0.139·12-s − 5.04·13-s − 1.46·14-s − 4.25·16-s + 6.36·17-s − 1.46·18-s − 5.32·19-s − 21-s + 1.46·22-s − 4.92·23-s − 2.72·24-s + 7.37·26-s − 27-s + 0.139·28-s + 5.04·29-s − 7.57·31-s + 0.786·32-s + 33-s − 9.31·34-s + ⋯ |
| L(s) = 1 | − 1.03·2-s − 0.577·3-s + 0.0695·4-s + 0.597·6-s + 0.377·7-s + 0.962·8-s + 0.333·9-s − 0.301·11-s − 0.0401·12-s − 1.39·13-s − 0.390·14-s − 1.06·16-s + 1.54·17-s − 0.344·18-s − 1.22·19-s − 0.218·21-s + 0.311·22-s − 1.02·23-s − 0.555·24-s + 1.44·26-s − 0.192·27-s + 0.0263·28-s + 0.936·29-s − 1.35·31-s + 0.138·32-s + 0.174·33-s − 1.59·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5017385026\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5017385026\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 1.46T + 2T^{2} \) |
| 13 | \( 1 + 5.04T + 13T^{2} \) |
| 17 | \( 1 - 6.36T + 17T^{2} \) |
| 19 | \( 1 + 5.32T + 19T^{2} \) |
| 23 | \( 1 + 4.92T + 23T^{2} \) |
| 29 | \( 1 - 5.04T + 29T^{2} \) |
| 31 | \( 1 + 7.57T + 31T^{2} \) |
| 37 | \( 1 + 4.24T + 37T^{2} \) |
| 41 | \( 1 + 0.646T + 41T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 + 0.526T + 47T^{2} \) |
| 53 | \( 1 + 3.72T + 53T^{2} \) |
| 59 | \( 1 - 7.97T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 8.76T + 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 - 13.0T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 + 13.1T + 83T^{2} \) |
| 89 | \( 1 - 11.8T + 89T^{2} \) |
| 97 | \( 1 - 1.87T + 97T^{2} \) |
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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
−7.991114457878502329051201751002, −7.62828275002694268921032578958, −6.96810165013271856414379147064, −5.97192293392954886811664323102, −5.22056707018586083229444120762, −4.60907806443572559478852371693, −3.78828461773882395978268917429, −2.44782139266176723191044398955, −1.60859633750760873327966472775, −0.46212839077164552854046041377,
0.46212839077164552854046041377, 1.60859633750760873327966472775, 2.44782139266176723191044398955, 3.78828461773882395978268917429, 4.60907806443572559478852371693, 5.22056707018586083229444120762, 5.97192293392954886811664323102, 6.96810165013271856414379147064, 7.62828275002694268921032578958, 7.991114457878502329051201751002
