| L(s) = 1 | + 2.57·2-s − 0.536·3-s + 4.62·4-s − 2.52·5-s − 1.38·6-s + 6.76·8-s − 2.71·9-s − 6.49·10-s + 5.05·11-s − 2.48·12-s + 2.83·13-s + 1.35·15-s + 8.16·16-s + 3.86·17-s − 6.98·18-s + 5.06·19-s − 11.6·20-s + 13.0·22-s + 5.17·23-s − 3.62·24-s + 1.35·25-s + 7.30·26-s + 3.06·27-s − 2.48·29-s + 3.48·30-s + 31-s + 7.48·32-s + ⋯ |
| L(s) = 1 | + 1.82·2-s − 0.309·3-s + 2.31·4-s − 1.12·5-s − 0.563·6-s + 2.39·8-s − 0.904·9-s − 2.05·10-s + 1.52·11-s − 0.716·12-s + 0.787·13-s + 0.349·15-s + 2.04·16-s + 0.936·17-s − 1.64·18-s + 1.16·19-s − 2.60·20-s + 2.77·22-s + 1.07·23-s − 0.740·24-s + 0.271·25-s + 1.43·26-s + 0.589·27-s − 0.461·29-s + 0.635·30-s + 0.179·31-s + 1.32·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1519 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1519 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.266008327\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.266008327\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 - 2.57T + 2T^{2} \) |
| 3 | \( 1 + 0.536T + 3T^{2} \) |
| 5 | \( 1 + 2.52T + 5T^{2} \) |
| 11 | \( 1 - 5.05T + 11T^{2} \) |
| 13 | \( 1 - 2.83T + 13T^{2} \) |
| 17 | \( 1 - 3.86T + 17T^{2} \) |
| 19 | \( 1 - 5.06T + 19T^{2} \) |
| 23 | \( 1 - 5.17T + 23T^{2} \) |
| 29 | \( 1 + 2.48T + 29T^{2} \) |
| 37 | \( 1 + 6.27T + 37T^{2} \) |
| 41 | \( 1 - 2.36T + 41T^{2} \) |
| 43 | \( 1 - 9.59T + 43T^{2} \) |
| 47 | \( 1 + 9.98T + 47T^{2} \) |
| 53 | \( 1 + 8.60T + 53T^{2} \) |
| 59 | \( 1 - 13.3T + 59T^{2} \) |
| 61 | \( 1 + 3.44T + 61T^{2} \) |
| 67 | \( 1 + 8.35T + 67T^{2} \) |
| 71 | \( 1 - 0.885T + 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 - 7.15T + 79T^{2} \) |
| 83 | \( 1 + 2.67T + 83T^{2} \) |
| 89 | \( 1 + 7.07T + 89T^{2} \) |
| 97 | \( 1 + 3.20T + 97T^{2} \) |
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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
−9.479270413241999523118933678124, −8.478724732108209775337007595758, −7.49966828860248964203729085392, −6.79397303899326625547280893083, −5.95599339576632132627300498336, −5.30323468997315811150769530181, −4.31985168517909305475304727328, −3.54381546568177010766617762009, −3.06735341933858476966165172878, −1.28265972808765726101856927457,
1.28265972808765726101856927457, 3.06735341933858476966165172878, 3.54381546568177010766617762009, 4.31985168517909305475304727328, 5.30323468997315811150769530181, 5.95599339576632132627300498336, 6.79397303899326625547280893083, 7.49966828860248964203729085392, 8.478724732108209775337007595758, 9.479270413241999523118933678124
