| L(s) = 1 | + 1.11·2-s − 2.45·3-s − 0.765·4-s − 3.92·5-s − 2.73·6-s − 0.931·7-s − 3.07·8-s + 3.03·9-s − 4.35·10-s + 1.51·11-s + 1.88·12-s − 3.02·13-s − 1.03·14-s + 9.63·15-s − 1.88·16-s − 6.39·17-s + 3.37·18-s + 5.06·19-s + 3.00·20-s + 2.28·21-s + 1.67·22-s + 3.08·23-s + 7.55·24-s + 10.3·25-s − 3.35·26-s − 0.0937·27-s + 0.712·28-s + ⋯ |
| L(s) = 1 | + 0.785·2-s − 1.41·3-s − 0.382·4-s − 1.75·5-s − 1.11·6-s − 0.351·7-s − 1.08·8-s + 1.01·9-s − 1.37·10-s + 0.455·11-s + 0.542·12-s − 0.838·13-s − 0.276·14-s + 2.48·15-s − 0.470·16-s − 1.55·17-s + 0.795·18-s + 1.16·19-s + 0.671·20-s + 0.499·21-s + 0.357·22-s + 0.644·23-s + 1.54·24-s + 2.07·25-s − 0.658·26-s − 0.0180·27-s + 0.134·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3970702231\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3970702231\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 859 | \( 1 - T \) |
| good | 2 | \( 1 - 1.11T + 2T^{2} \) |
| 3 | \( 1 + 2.45T + 3T^{2} \) |
| 5 | \( 1 + 3.92T + 5T^{2} \) |
| 7 | \( 1 + 0.931T + 7T^{2} \) |
| 11 | \( 1 - 1.51T + 11T^{2} \) |
| 13 | \( 1 + 3.02T + 13T^{2} \) |
| 17 | \( 1 + 6.39T + 17T^{2} \) |
| 19 | \( 1 - 5.06T + 19T^{2} \) |
| 23 | \( 1 - 3.08T + 23T^{2} \) |
| 29 | \( 1 - 6.09T + 29T^{2} \) |
| 31 | \( 1 + 9.17T + 31T^{2} \) |
| 37 | \( 1 + 10.5T + 37T^{2} \) |
| 41 | \( 1 - 1.68T + 41T^{2} \) |
| 43 | \( 1 - 4.17T + 43T^{2} \) |
| 47 | \( 1 - 1.11T + 47T^{2} \) |
| 53 | \( 1 - 4.53T + 53T^{2} \) |
| 59 | \( 1 + 3.90T + 59T^{2} \) |
| 61 | \( 1 - 13.2T + 61T^{2} \) |
| 67 | \( 1 + 3.67T + 67T^{2} \) |
| 71 | \( 1 - 8.70T + 71T^{2} \) |
| 73 | \( 1 + 5.56T + 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 + 6.38T + 83T^{2} \) |
| 89 | \( 1 - 6.60T + 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
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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
−10.52781646512778671242992961216, −9.297243989496171822918267654864, −8.510100942883670229807380576501, −7.18293187289229986259641781634, −6.74395096455662647287964196066, −5.49735826858445365881319162483, −4.79030940304249304285044517633, −4.11403808478721394495886393039, −3.14432399627151291264169037005, −0.46533194415419191852389522859,
0.46533194415419191852389522859, 3.14432399627151291264169037005, 4.11403808478721394495886393039, 4.79030940304249304285044517633, 5.49735826858445365881319162483, 6.74395096455662647287964196066, 7.18293187289229986259641781634, 8.510100942883670229807380576501, 9.297243989496171822918267654864, 10.52781646512778671242992961216
