| L(s) = 1 | − 0.937·2-s + 1.68·3-s − 1.12·4-s + 0.290·5-s − 1.58·6-s − 3.85·7-s + 2.92·8-s − 0.158·9-s − 0.272·10-s + 2.09·11-s − 1.89·12-s + 3.61·14-s + 0.490·15-s − 0.499·16-s − 5.29·17-s + 0.148·18-s + 3.22·19-s − 0.326·20-s − 6.49·21-s − 1.96·22-s + 6.80·23-s + 4.93·24-s − 4.91·25-s − 5.32·27-s + 4.32·28-s − 4.52·29-s − 0.459·30-s + ⋯ |
| L(s) = 1 | − 0.662·2-s + 0.973·3-s − 0.560·4-s + 0.130·5-s − 0.645·6-s − 1.45·7-s + 1.03·8-s − 0.0527·9-s − 0.0861·10-s + 0.632·11-s − 0.545·12-s + 0.965·14-s + 0.126·15-s − 0.124·16-s − 1.28·17-s + 0.0349·18-s + 0.739·19-s − 0.0729·20-s − 1.41·21-s − 0.418·22-s + 1.41·23-s + 1.00·24-s − 0.983·25-s − 1.02·27-s + 0.817·28-s − 0.839·29-s − 0.0838·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.045962240\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.045962240\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 0.937T + 2T^{2} \) |
| 3 | \( 1 - 1.68T + 3T^{2} \) |
| 5 | \( 1 - 0.290T + 5T^{2} \) |
| 7 | \( 1 + 3.85T + 7T^{2} \) |
| 11 | \( 1 - 2.09T + 11T^{2} \) |
| 17 | \( 1 + 5.29T + 17T^{2} \) |
| 19 | \( 1 - 3.22T + 19T^{2} \) |
| 23 | \( 1 - 6.80T + 23T^{2} \) |
| 29 | \( 1 + 4.52T + 29T^{2} \) |
| 37 | \( 1 + 4.81T + 37T^{2} \) |
| 41 | \( 1 - 6.15T + 41T^{2} \) |
| 43 | \( 1 - 7.12T + 43T^{2} \) |
| 47 | \( 1 - 4.76T + 47T^{2} \) |
| 53 | \( 1 + 5.08T + 53T^{2} \) |
| 59 | \( 1 - 9.83T + 59T^{2} \) |
| 61 | \( 1 + 9.60T + 61T^{2} \) |
| 67 | \( 1 - 6.38T + 67T^{2} \) |
| 71 | \( 1 + 3.60T + 71T^{2} \) |
| 73 | \( 1 - 8.21T + 73T^{2} \) |
| 79 | \( 1 + 4.16T + 79T^{2} \) |
| 83 | \( 1 - 5.60T + 83T^{2} \) |
| 89 | \( 1 + 5.99T + 89T^{2} \) |
| 97 | \( 1 - 1.49T + 97T^{2} \) |
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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
−8.480654974450810141846954310871, −7.52740213765595372627964850955, −7.06062738537923271199745220766, −6.15672867444504061019236361162, −5.35618697064601752917841991650, −4.22692846806144473934000496653, −3.62767065795613799235363578592, −2.86687565897353448322158349543, −1.90035168632615106763777316154, −0.57256745213532375072332674512,
0.57256745213532375072332674512, 1.90035168632615106763777316154, 2.86687565897353448322158349543, 3.62767065795613799235363578592, 4.22692846806144473934000496653, 5.35618697064601752917841991650, 6.15672867444504061019236361162, 7.06062738537923271199745220766, 7.52740213765595372627964850955, 8.480654974450810141846954310871
