L(s) = 1 | − 7.95·2-s + 14.0·3-s + 31.2·4-s + 25·5-s − 111.·6-s + 100.·7-s + 6.27·8-s − 45.7·9-s − 198.·10-s + 121·11-s + 438.·12-s + 570.·13-s − 801.·14-s + 351.·15-s − 1.04e3·16-s − 1.18e3·17-s + 363.·18-s − 361·19-s + 780.·20-s + 1.41e3·21-s − 962.·22-s + 3.60e3·23-s + 88.1·24-s + 625·25-s − 4.53e3·26-s − 4.05e3·27-s + 3.14e3·28-s + ⋯ |
L(s) = 1 | − 1.40·2-s + 0.900·3-s + 0.975·4-s + 0.447·5-s − 1.26·6-s + 0.777·7-s + 0.0346·8-s − 0.188·9-s − 0.628·10-s + 0.301·11-s + 0.878·12-s + 0.936·13-s − 1.09·14-s + 0.402·15-s − 1.02·16-s − 0.990·17-s + 0.264·18-s − 0.229·19-s + 0.436·20-s + 0.700·21-s − 0.423·22-s + 1.42·23-s + 0.0312·24-s + 0.200·25-s − 1.31·26-s − 1.07·27-s + 0.758·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 + 361T \) |
good | 2 | \( 1 + 7.95T + 32T^{2} \) |
| 3 | \( 1 - 14.0T + 243T^{2} \) |
| 7 | \( 1 - 100.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 570.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.18e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 3.60e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.10e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.96e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.36e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.14e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.60e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 617.T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.08e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.42e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 8.29e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.14e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.69e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.89e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.12e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.69e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.48e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.70e5T + 8.58e9T^{2} \) |
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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
−8.751918337656831252595771186909, −8.396781280320595336439422724807, −7.39289248212774971659398149406, −6.64767768385755558402614835680, −5.41400178641222098222336631341, −4.27277328248998444186851448953, −3.05086100793223957167795583032, −1.94102796326827207030161141828, −1.35110140372277150722634477740, 0,
1.35110140372277150722634477740, 1.94102796326827207030161141828, 3.05086100793223957167795583032, 4.27277328248998444186851448953, 5.41400178641222098222336631341, 6.64767768385755558402614835680, 7.39289248212774971659398149406, 8.396781280320595336439422724807, 8.751918337656831252595771186909