| L(s) = 1 | − 4.51·2-s + 3·3-s + 12.4·4-s + 19.2·5-s − 13.5·6-s − 8.39·7-s − 19.8·8-s + 9·9-s − 87.0·10-s + 11·11-s + 37.2·12-s − 69.5·13-s + 37.9·14-s + 57.8·15-s − 9.36·16-s + 25.9·17-s − 40.6·18-s − 84.4·19-s + 239.·20-s − 25.1·21-s − 49.6·22-s − 153.·23-s − 59.6·24-s + 246.·25-s + 314.·26-s + 27·27-s − 104.·28-s + ⋯ |
| L(s) = 1 | − 1.59·2-s + 0.577·3-s + 1.55·4-s + 1.72·5-s − 0.922·6-s − 0.453·7-s − 0.879·8-s + 0.333·9-s − 2.75·10-s + 0.301·11-s + 0.895·12-s − 1.48·13-s + 0.723·14-s + 0.995·15-s − 0.146·16-s + 0.370·17-s − 0.532·18-s − 1.01·19-s + 2.67·20-s − 0.261·21-s − 0.481·22-s − 1.39·23-s − 0.507·24-s + 1.97·25-s + 2.36·26-s + 0.192·27-s − 0.702·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 11 | \( 1 - 11T \) |
| 61 | \( 1 - 61T \) |
| good | 2 | \( 1 + 4.51T + 8T^{2} \) |
| 5 | \( 1 - 19.2T + 125T^{2} \) |
| 7 | \( 1 + 8.39T + 343T^{2} \) |
| 13 | \( 1 + 69.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 25.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 84.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 153.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 105.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 175.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 295.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 381.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 516.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 310.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 535.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 319.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 663.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 601.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 141.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.15e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 313.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.29e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 751.T + 9.12e5T^{2} \) |
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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
−8.597458629222925489747912598910, −7.85810789061147468620857303470, −6.96331469517341845837148237770, −6.37141686776650337916718512729, −5.47507398155251156254601948236, −4.25307174513779016550502952766, −2.63376693347067963571805351615, −2.21859712154347064820535034255, −1.32000485672655506379859156833, 0,
1.32000485672655506379859156833, 2.21859712154347064820535034255, 2.63376693347067963571805351615, 4.25307174513779016550502952766, 5.47507398155251156254601948236, 6.37141686776650337916718512729, 6.96331469517341845837148237770, 7.85810789061147468620857303470, 8.597458629222925489747912598910
