| L(s) = 1 | + 2-s − 2·3-s + 4-s − 2.73·5-s − 2·6-s + 8-s + 9-s − 2.73·10-s − 3.46·11-s − 2·12-s + 2.73·13-s + 5.46·15-s + 16-s + 6.19·17-s + 18-s + 3.46·19-s − 2.73·20-s − 3.46·22-s + 4.92·23-s − 2·24-s + 2.46·25-s + 2.73·26-s + 4·27-s − 29-s + 5.46·30-s − 8.19·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 0.5·4-s − 1.22·5-s − 0.816·6-s + 0.353·8-s + 0.333·9-s − 0.863·10-s − 1.04·11-s − 0.577·12-s + 0.757·13-s + 1.41·15-s + 0.250·16-s + 1.50·17-s + 0.235·18-s + 0.794·19-s − 0.610·20-s − 0.738·22-s + 1.02·23-s − 0.408·24-s + 0.492·25-s + 0.535·26-s + 0.769·27-s − 0.185·29-s + 0.997·30-s − 1.47·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 + 2T + 3T^{2} \) |
| 5 | \( 1 + 2.73T + 5T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 - 2.73T + 13T^{2} \) |
| 17 | \( 1 - 6.19T + 17T^{2} \) |
| 19 | \( 1 - 3.46T + 19T^{2} \) |
| 23 | \( 1 - 4.92T + 23T^{2} \) |
| 31 | \( 1 + 8.19T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + 7.66T + 41T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 + 9.66T + 47T^{2} \) |
| 53 | \( 1 + 8.92T + 53T^{2} \) |
| 59 | \( 1 + 7.26T + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 - 6.92T + 67T^{2} \) |
| 71 | \( 1 + 4.92T + 71T^{2} \) |
| 73 | \( 1 + 8.73T + 73T^{2} \) |
| 79 | \( 1 + 9.46T + 79T^{2} \) |
| 83 | \( 1 - 14.5T + 83T^{2} \) |
| 89 | \( 1 + 5.80T + 89T^{2} \) |
| 97 | \( 1 - 3.26T + 97T^{2} \) |
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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
−8.048820929375021945754628518472, −7.58505852444811383056955677834, −6.83403409000343597068862487034, −5.81868690948606170480661028541, −5.33625844595944660158278373400, −4.68464308557591876629991324699, −3.55475410255333249863939777162, −3.07726971834435539859597113764, −1.29922312819189031317424962441, 0,
1.29922312819189031317424962441, 3.07726971834435539859597113764, 3.55475410255333249863939777162, 4.68464308557591876629991324699, 5.33625844595944660158278373400, 5.81868690948606170480661028541, 6.83403409000343597068862487034, 7.58505852444811383056955677834, 8.048820929375021945754628518472
