| L(s) = 1 | − 4.89·2-s + 7.57·3-s + 15.9·4-s − 14.0·5-s − 37.1·6-s + 7·7-s − 39.1·8-s + 30.4·9-s + 68.6·10-s + 35.1·11-s + 121.·12-s + 16.8·13-s − 34.2·14-s − 106.·15-s + 63.6·16-s − 71.4·17-s − 149.·18-s + 133.·19-s − 224.·20-s + 53.0·21-s − 172.·22-s + 23·23-s − 296.·24-s + 71.5·25-s − 82.5·26-s + 26.1·27-s + 111.·28-s + ⋯ |
| L(s) = 1 | − 1.73·2-s + 1.45·3-s + 1.99·4-s − 1.25·5-s − 2.52·6-s + 0.377·7-s − 1.72·8-s + 1.12·9-s + 2.17·10-s + 0.964·11-s + 2.91·12-s + 0.359·13-s − 0.654·14-s − 1.82·15-s + 0.995·16-s − 1.01·17-s − 1.95·18-s + 1.61·19-s − 2.50·20-s + 0.551·21-s − 1.66·22-s + 0.208·23-s − 2.52·24-s + 0.572·25-s − 0.622·26-s + 0.186·27-s + 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.104989260\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.104989260\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 7T \) |
| 23 | \( 1 - 23T \) |
| good | 2 | \( 1 + 4.89T + 8T^{2} \) |
| 3 | \( 1 - 7.57T + 27T^{2} \) |
| 5 | \( 1 + 14.0T + 125T^{2} \) |
| 11 | \( 1 - 35.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 16.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 71.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 133.T + 6.85e3T^{2} \) |
| 29 | \( 1 - 240.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 105.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 329.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 304.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 140.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 298.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 292.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 62.9T + 2.05e5T^{2} \) |
| 61 | \( 1 - 593.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 744.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 277.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 551.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 929.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 749.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.12e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.46e3T + 9.12e5T^{2} \) |
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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
−11.87542680904860155354485847860, −11.32048202586634615096267482615, −9.982615059234090070294019109302, −9.011519261159110180393497374774, −8.423527456025767877324361339502, −7.69737196173257455539542391650, −6.83006638849255927392028746055, −4.09727562237209250355203207818, −2.73612913402671256650040092845, −1.08771885479055495262375618669,
1.08771885479055495262375618669, 2.73612913402671256650040092845, 4.09727562237209250355203207818, 6.83006638849255927392028746055, 7.69737196173257455539542391650, 8.423527456025767877324361339502, 9.011519261159110180393497374774, 9.982615059234090070294019109302, 11.32048202586634615096267482615, 11.87542680904860155354485847860
