| L(s) = 1 | + 2.46·2-s + 3.39·3-s + 4.08·4-s − 1.74·5-s + 8.36·6-s − 7-s + 5.12·8-s + 8.49·9-s − 4.30·10-s + 0.0793·11-s + 13.8·12-s − 2.97·13-s − 2.46·14-s − 5.91·15-s + 4.48·16-s + 3.54·17-s + 20.9·18-s − 5.22·19-s − 7.12·20-s − 3.39·21-s + 0.195·22-s + 0.894·23-s + 17.3·24-s − 1.95·25-s − 7.34·26-s + 18.6·27-s − 4.08·28-s + ⋯ |
| L(s) = 1 | + 1.74·2-s + 1.95·3-s + 2.04·4-s − 0.780·5-s + 3.41·6-s − 0.377·7-s + 1.81·8-s + 2.83·9-s − 1.36·10-s + 0.0239·11-s + 3.99·12-s − 0.826·13-s − 0.659·14-s − 1.52·15-s + 1.12·16-s + 0.859·17-s + 4.93·18-s − 1.19·19-s − 1.59·20-s − 0.739·21-s + 0.0417·22-s + 0.186·23-s + 3.55·24-s − 0.390·25-s − 1.44·26-s + 3.58·27-s − 0.771·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1589 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1589 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.665681218\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.665681218\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 227 | \( 1 - T \) |
| good | 2 | \( 1 - 2.46T + 2T^{2} \) |
| 3 | \( 1 - 3.39T + 3T^{2} \) |
| 5 | \( 1 + 1.74T + 5T^{2} \) |
| 11 | \( 1 - 0.0793T + 11T^{2} \) |
| 13 | \( 1 + 2.97T + 13T^{2} \) |
| 17 | \( 1 - 3.54T + 17T^{2} \) |
| 19 | \( 1 + 5.22T + 19T^{2} \) |
| 23 | \( 1 - 0.894T + 23T^{2} \) |
| 29 | \( 1 + 4.02T + 29T^{2} \) |
| 31 | \( 1 + 6.38T + 31T^{2} \) |
| 37 | \( 1 - 8.66T + 37T^{2} \) |
| 41 | \( 1 + 2.54T + 41T^{2} \) |
| 43 | \( 1 - 5.35T + 43T^{2} \) |
| 47 | \( 1 + 8.27T + 47T^{2} \) |
| 53 | \( 1 - 4.27T + 53T^{2} \) |
| 59 | \( 1 - 6.38T + 59T^{2} \) |
| 61 | \( 1 + 0.414T + 61T^{2} \) |
| 67 | \( 1 + 6.97T + 67T^{2} \) |
| 71 | \( 1 + 5.51T + 71T^{2} \) |
| 73 | \( 1 - 1.64T + 73T^{2} \) |
| 79 | \( 1 - 7.45T + 79T^{2} \) |
| 83 | \( 1 - 5.16T + 83T^{2} \) |
| 89 | \( 1 + 10.7T + 89T^{2} \) |
| 97 | \( 1 + 4.73T + 97T^{2} \) |
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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
−9.370525193679757821871872479456, −8.434520232635952359201077192719, −7.53475007934648487606582842126, −7.22398646136383377296692676454, −6.12215837486950144946206764202, −4.86093894784998764436860941128, −4.04575773560956445173698980840, −3.58058476630324112748351081430, −2.75594510034076364976392047117, −1.95751821386776681090966656436,
1.95751821386776681090966656436, 2.75594510034076364976392047117, 3.58058476630324112748351081430, 4.04575773560956445173698980840, 4.86093894784998764436860941128, 6.12215837486950144946206764202, 7.22398646136383377296692676454, 7.53475007934648487606582842126, 8.434520232635952359201077192719, 9.370525193679757821871872479456
