| L(s) = 1 | + 2.71·2-s + 0.265·3-s + 5.36·4-s − 0.954·5-s + 0.721·6-s + 4.28·7-s + 9.13·8-s − 2.92·9-s − 2.58·10-s + 0.0225·11-s + 1.42·12-s + 11.6·14-s − 0.253·15-s + 14.0·16-s + 4.73·17-s − 7.94·18-s + 1.51·19-s − 5.11·20-s + 1.13·21-s + 0.0610·22-s + 7.63·23-s + 2.42·24-s − 4.08·25-s − 1.57·27-s + 22.9·28-s − 2.41·29-s − 0.688·30-s + ⋯ |
| L(s) = 1 | + 1.91·2-s + 0.153·3-s + 2.68·4-s − 0.426·5-s + 0.294·6-s + 1.61·7-s + 3.22·8-s − 0.976·9-s − 0.818·10-s + 0.00678·11-s + 0.411·12-s + 3.10·14-s − 0.0654·15-s + 3.51·16-s + 1.14·17-s − 1.87·18-s + 0.348·19-s − 1.14·20-s + 0.248·21-s + 0.0130·22-s + 1.59·23-s + 0.495·24-s − 0.817·25-s − 0.303·27-s + 4.34·28-s − 0.448·29-s − 0.125·30-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
| \(L(1)\) |
\(\approx\) |
\(8.501859893\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.501859893\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.71T + 2T^{2} \) |
| 3 | \( 1 - 0.265T + 3T^{2} \) |
| 5 | \( 1 + 0.954T + 5T^{2} \) |
| 7 | \( 1 - 4.28T + 7T^{2} \) |
| 11 | \( 1 - 0.0225T + 11T^{2} \) |
| 17 | \( 1 - 4.73T + 17T^{2} \) |
| 19 | \( 1 - 1.51T + 19T^{2} \) |
| 23 | \( 1 - 7.63T + 23T^{2} \) |
| 29 | \( 1 + 2.41T + 29T^{2} \) |
| 37 | \( 1 + 2.46T + 37T^{2} \) |
| 41 | \( 1 + 6.00T + 41T^{2} \) |
| 43 | \( 1 + 9.18T + 43T^{2} \) |
| 47 | \( 1 + 6.05T + 47T^{2} \) |
| 53 | \( 1 + 5.67T + 53T^{2} \) |
| 59 | \( 1 - 8.26T + 59T^{2} \) |
| 61 | \( 1 - 9.28T + 61T^{2} \) |
| 67 | \( 1 + 7.52T + 67T^{2} \) |
| 71 | \( 1 - 13.2T + 71T^{2} \) |
| 73 | \( 1 - 1.76T + 73T^{2} \) |
| 79 | \( 1 - 10.0T + 79T^{2} \) |
| 83 | \( 1 + 13.3T + 83T^{2} \) |
| 89 | \( 1 - 6.66T + 89T^{2} \) |
| 97 | \( 1 + 10.9T + 97T^{2} \) |
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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
−8.021905026287502150490946193254, −7.34144974362087068507297046575, −6.63670400322419750027467530994, −5.54810738325620367005097344467, −5.25812552641003295853382781650, −4.69228823407447375256167764857, −3.65824580068594827778947694810, −3.21805402199525423766285066216, −2.20035920546710841420798156334, −1.34863898605134952484755061493,
1.34863898605134952484755061493, 2.20035920546710841420798156334, 3.21805402199525423766285066216, 3.65824580068594827778947694810, 4.69228823407447375256167764857, 5.25812552641003295853382781650, 5.54810738325620367005097344467, 6.63670400322419750027467530994, 7.34144974362087068507297046575, 8.021905026287502150490946193254
