| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 3.96·7-s − 8-s + 9-s + 10-s + 3.04·11-s − 12-s + 0.919·13-s + 3.96·14-s + 15-s + 16-s + 0.351·17-s − 18-s − 0.351·19-s − 20-s + 3.96·21-s − 3.04·22-s − 3.43·23-s + 24-s + 25-s − 0.919·26-s − 27-s − 3.96·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s + 0.408·6-s − 1.50·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.919·11-s − 0.288·12-s + 0.254·13-s + 1.06·14-s + 0.258·15-s + 0.250·16-s + 0.0853·17-s − 0.235·18-s − 0.0807·19-s − 0.223·20-s + 0.866·21-s − 0.650·22-s − 0.715·23-s + 0.204·24-s + 0.200·25-s − 0.180·26-s − 0.192·27-s − 0.750·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 + T \) |
| good | 7 | \( 1 + 3.96T + 7T^{2} \) |
| 11 | \( 1 - 3.04T + 11T^{2} \) |
| 13 | \( 1 - 0.919T + 13T^{2} \) |
| 17 | \( 1 - 0.351T + 17T^{2} \) |
| 19 | \( 1 + 0.351T + 19T^{2} \) |
| 23 | \( 1 + 3.43T + 23T^{2} \) |
| 29 | \( 1 - 9.37T + 29T^{2} \) |
| 31 | \( 1 - 2.91T + 31T^{2} \) |
| 37 | \( 1 - 5.40T + 37T^{2} \) |
| 41 | \( 1 + 8.45T + 41T^{2} \) |
| 43 | \( 1 - 6.09T + 43T^{2} \) |
| 47 | \( 1 + 13.2T + 47T^{2} \) |
| 53 | \( 1 - 5.93T + 53T^{2} \) |
| 59 | \( 1 + 8.45T + 59T^{2} \) |
| 61 | \( 1 + 1.96T + 61T^{2} \) |
| 71 | \( 1 + 5.61T + 71T^{2} \) |
| 73 | \( 1 + 12.8T + 73T^{2} \) |
| 79 | \( 1 - 11.8T + 79T^{2} \) |
| 83 | \( 1 - 0.105T + 83T^{2} \) |
| 89 | \( 1 + 16.2T + 89T^{2} \) |
| 97 | \( 1 - 3.04T + 97T^{2} \) |
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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
−8.823912049941570337962900435899, −8.069963344674708502874829395133, −7.04638203403860069670856862366, −6.44282755517497983071593833527, −6.00133534193021209741264634100, −4.61992845920438527234534832753, −3.67344504429635174646239847932, −2.79659753029482289526508617421, −1.22579544315174919631868493772, 0,
1.22579544315174919631868493772, 2.79659753029482289526508617421, 3.67344504429635174646239847932, 4.61992845920438527234534832753, 6.00133534193021209741264634100, 6.44282755517497983071593833527, 7.04638203403860069670856862366, 8.069963344674708502874829395133, 8.823912049941570337962900435899
