L(s) = 1 | − 2-s + 1.55·3-s + 4-s − 1.55·5-s − 1.55·6-s + 1.46·7-s − 8-s − 0.586·9-s + 1.55·10-s + 1.61·11-s + 1.55·12-s − 1.39·13-s − 1.46·14-s − 2.41·15-s + 16-s − 0.623·17-s + 0.586·18-s + 1.68·19-s − 1.55·20-s + 2.27·21-s − 1.61·22-s + 1.66·23-s − 1.55·24-s − 2.58·25-s + 1.39·26-s − 5.57·27-s + 1.46·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.896·3-s + 0.5·4-s − 0.695·5-s − 0.634·6-s + 0.553·7-s − 0.353·8-s − 0.195·9-s + 0.491·10-s + 0.488·11-s + 0.448·12-s − 0.388·13-s − 0.391·14-s − 0.623·15-s + 0.250·16-s − 0.151·17-s + 0.138·18-s + 0.385·19-s − 0.347·20-s + 0.496·21-s − 0.345·22-s + 0.347·23-s − 0.317·24-s − 0.516·25-s + 0.274·26-s − 1.07·27-s + 0.276·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 2003 | \( 1 + T \) |
good | 3 | \( 1 - 1.55T + 3T^{2} \) |
| 5 | \( 1 + 1.55T + 5T^{2} \) |
| 7 | \( 1 - 1.46T + 7T^{2} \) |
| 11 | \( 1 - 1.61T + 11T^{2} \) |
| 13 | \( 1 + 1.39T + 13T^{2} \) |
| 17 | \( 1 + 0.623T + 17T^{2} \) |
| 19 | \( 1 - 1.68T + 19T^{2} \) |
| 23 | \( 1 - 1.66T + 23T^{2} \) |
| 29 | \( 1 + 10.3T + 29T^{2} \) |
| 31 | \( 1 - 2.33T + 31T^{2} \) |
| 37 | \( 1 - 2.86T + 37T^{2} \) |
| 41 | \( 1 - 5.35T + 41T^{2} \) |
| 43 | \( 1 + 4.21T + 43T^{2} \) |
| 47 | \( 1 + 8.95T + 47T^{2} \) |
| 53 | \( 1 - 1.11T + 53T^{2} \) |
| 59 | \( 1 - 7.62T + 59T^{2} \) |
| 61 | \( 1 + 13.6T + 61T^{2} \) |
| 67 | \( 1 - 5.97T + 67T^{2} \) |
| 71 | \( 1 + 8.75T + 71T^{2} \) |
| 73 | \( 1 - 10.5T + 73T^{2} \) |
| 79 | \( 1 - 8.36T + 79T^{2} \) |
| 83 | \( 1 + 16.3T + 83T^{2} \) |
| 89 | \( 1 - 3.11T + 89T^{2} \) |
| 97 | \( 1 - 13.4T + 97T^{2} \) |
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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
−7.937998125078021724355511662800, −7.77919315761363982422102879745, −6.93875444082635784858646455477, −5.97782914536266725465812626877, −5.04777489551949568944961138408, −4.01141568637446817103256055027, −3.33077281647394750874332692785, −2.39198856640236127526460743997, −1.48858235303623212569325405459, 0,
1.48858235303623212569325405459, 2.39198856640236127526460743997, 3.33077281647394750874332692785, 4.01141568637446817103256055027, 5.04777489551949568944961138408, 5.97782914536266725465812626877, 6.93875444082635784858646455477, 7.77919315761363982422102879745, 7.937998125078021724355511662800