L(s) = 1 | − 0.0717·2-s − 3-s − 1.99·4-s − 1.92·5-s + 0.0717·6-s + 3.89·7-s + 0.286·8-s + 9-s + 0.137·10-s + 0.330·11-s + 1.99·12-s + 1.95·13-s − 0.279·14-s + 1.92·15-s + 3.96·16-s − 17-s − 0.0717·18-s + 7.45·19-s + 3.83·20-s − 3.89·21-s − 0.0236·22-s − 0.344·23-s − 0.286·24-s − 1.29·25-s − 0.140·26-s − 27-s − 7.76·28-s + ⋯ |
L(s) = 1 | − 0.0507·2-s − 0.577·3-s − 0.997·4-s − 0.860·5-s + 0.0292·6-s + 1.47·7-s + 0.101·8-s + 0.333·9-s + 0.0436·10-s + 0.0995·11-s + 0.575·12-s + 0.542·13-s − 0.0746·14-s + 0.496·15-s + 0.992·16-s − 0.242·17-s − 0.0169·18-s + 1.71·19-s + 0.858·20-s − 0.849·21-s − 0.00505·22-s − 0.0718·23-s − 0.0584·24-s − 0.259·25-s − 0.0275·26-s − 0.192·27-s − 1.46·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.203170637\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.203170637\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 79 | \( 1 - T \) |
good | 2 | \( 1 + 0.0717T + 2T^{2} \) |
| 5 | \( 1 + 1.92T + 5T^{2} \) |
| 7 | \( 1 - 3.89T + 7T^{2} \) |
| 11 | \( 1 - 0.330T + 11T^{2} \) |
| 13 | \( 1 - 1.95T + 13T^{2} \) |
| 19 | \( 1 - 7.45T + 19T^{2} \) |
| 23 | \( 1 + 0.344T + 23T^{2} \) |
| 29 | \( 1 + 3.99T + 29T^{2} \) |
| 31 | \( 1 - 6.60T + 31T^{2} \) |
| 37 | \( 1 + 0.669T + 37T^{2} \) |
| 41 | \( 1 + 3.05T + 41T^{2} \) |
| 43 | \( 1 - 4.13T + 43T^{2} \) |
| 47 | \( 1 - 3.35T + 47T^{2} \) |
| 53 | \( 1 - 10.2T + 53T^{2} \) |
| 59 | \( 1 + 14.3T + 59T^{2} \) |
| 61 | \( 1 - 9.85T + 61T^{2} \) |
| 67 | \( 1 + 7.02T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 + 13.8T + 73T^{2} \) |
| 83 | \( 1 + 13.5T + 83T^{2} \) |
| 89 | \( 1 - 14.9T + 89T^{2} \) |
| 97 | \( 1 + 8.01T + 97T^{2} \) |
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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
−8.466490649307748596057826243391, −7.61349604896222023274237739894, −7.38833878712373201970141703437, −5.96785290447495704884265685182, −5.37058885357714642777127013663, −4.58815938040811587331490738159, −4.14748082192480787886235201081, −3.20545539676812796442665590880, −1.59112939422867568310140697007, −0.71463169451144713378904167196,
0.71463169451144713378904167196, 1.59112939422867568310140697007, 3.20545539676812796442665590880, 4.14748082192480787886235201081, 4.58815938040811587331490738159, 5.37058885357714642777127013663, 5.96785290447495704884265685182, 7.38833878712373201970141703437, 7.61349604896222023274237739894, 8.466490649307748596057826243391