| L(s) = 1 | − 1.57·2-s − 3.26·3-s + 0.490·4-s + 1.75·5-s + 5.14·6-s − 4.57·7-s + 2.38·8-s + 7.64·9-s − 2.76·10-s − 3.30·11-s − 1.59·12-s − 4.85·13-s + 7.22·14-s − 5.72·15-s − 4.74·16-s + 5.91·17-s − 12.0·18-s + 5.00·19-s + 0.860·20-s + 14.9·21-s + 5.22·22-s − 6.05·23-s − 7.77·24-s − 1.92·25-s + 7.65·26-s − 15.1·27-s − 2.24·28-s + ⋯ |
| L(s) = 1 | − 1.11·2-s − 1.88·3-s + 0.245·4-s + 0.784·5-s + 2.10·6-s − 1.73·7-s + 0.842·8-s + 2.54·9-s − 0.875·10-s − 0.997·11-s − 0.461·12-s − 1.34·13-s + 1.93·14-s − 1.47·15-s − 1.18·16-s + 1.43·17-s − 2.84·18-s + 1.14·19-s + 0.192·20-s + 3.25·21-s + 1.11·22-s − 1.26·23-s − 1.58·24-s − 0.384·25-s + 1.50·26-s − 2.91·27-s − 0.424·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1266751357\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1266751357\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2671 | \( 1 - T \) |
| good | 2 | \( 1 + 1.57T + 2T^{2} \) |
| 3 | \( 1 + 3.26T + 3T^{2} \) |
| 5 | \( 1 - 1.75T + 5T^{2} \) |
| 7 | \( 1 + 4.57T + 7T^{2} \) |
| 11 | \( 1 + 3.30T + 11T^{2} \) |
| 13 | \( 1 + 4.85T + 13T^{2} \) |
| 17 | \( 1 - 5.91T + 17T^{2} \) |
| 19 | \( 1 - 5.00T + 19T^{2} \) |
| 23 | \( 1 + 6.05T + 23T^{2} \) |
| 29 | \( 1 + 1.14T + 29T^{2} \) |
| 31 | \( 1 - 0.334T + 31T^{2} \) |
| 37 | \( 1 - 5.34T + 37T^{2} \) |
| 41 | \( 1 + 3.22T + 41T^{2} \) |
| 43 | \( 1 - 5.43T + 43T^{2} \) |
| 47 | \( 1 + 4.51T + 47T^{2} \) |
| 53 | \( 1 + 14.1T + 53T^{2} \) |
| 59 | \( 1 + 5.94T + 59T^{2} \) |
| 61 | \( 1 + 9.41T + 61T^{2} \) |
| 67 | \( 1 - 3.98T + 67T^{2} \) |
| 71 | \( 1 + 1.79T + 71T^{2} \) |
| 73 | \( 1 + 8.95T + 73T^{2} \) |
| 79 | \( 1 + 15.7T + 79T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 - 16.1T + 89T^{2} \) |
| 97 | \( 1 + 16.7T + 97T^{2} \) |
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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
−9.556327521279504390245690629823, −7.75406918245434820652181596684, −7.44265277380322164320756134340, −6.50142863110487055856803209397, −5.78376517538986089812367899094, −5.32664949223724575445096059332, −4.34892216946902114906206938415, −2.94253008056321692228490582683, −1.54152888580035912430606079373, −0.29622950050553848219962141251,
0.29622950050553848219962141251, 1.54152888580035912430606079373, 2.94253008056321692228490582683, 4.34892216946902114906206938415, 5.32664949223724575445096059332, 5.78376517538986089812367899094, 6.50142863110487055856803209397, 7.44265277380322164320756134340, 7.75406918245434820652181596684, 9.556327521279504390245690629823
