| L(s) = 1 | − 0.695·2-s − 2.78·3-s − 1.51·4-s − 5-s + 1.93·6-s + 0.663·7-s + 2.44·8-s + 4.78·9-s + 0.695·10-s − 6.07·11-s + 4.23·12-s − 4.04·13-s − 0.461·14-s + 2.78·15-s + 1.33·16-s − 8.04·17-s − 3.32·18-s − 7.85·19-s + 1.51·20-s − 1.85·21-s + 4.22·22-s + 8.37·23-s − 6.81·24-s + 25-s + 2.80·26-s − 4.96·27-s − 1.00·28-s + ⋯ |
| L(s) = 1 | − 0.491·2-s − 1.61·3-s − 0.758·4-s − 0.447·5-s + 0.791·6-s + 0.250·7-s + 0.864·8-s + 1.59·9-s + 0.219·10-s − 1.83·11-s + 1.22·12-s − 1.12·13-s − 0.123·14-s + 0.720·15-s + 0.333·16-s − 1.95·17-s − 0.783·18-s − 1.80·19-s + 0.339·20-s − 0.403·21-s + 0.900·22-s + 1.74·23-s − 1.39·24-s + 0.200·25-s + 0.551·26-s − 0.955·27-s − 0.190·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 985 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 985 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.08554926856\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08554926856\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 197 | \( 1 - T \) |
| good | 2 | \( 1 + 0.695T + 2T^{2} \) |
| 3 | \( 1 + 2.78T + 3T^{2} \) |
| 7 | \( 1 - 0.663T + 7T^{2} \) |
| 11 | \( 1 + 6.07T + 11T^{2} \) |
| 13 | \( 1 + 4.04T + 13T^{2} \) |
| 17 | \( 1 + 8.04T + 17T^{2} \) |
| 19 | \( 1 + 7.85T + 19T^{2} \) |
| 23 | \( 1 - 8.37T + 23T^{2} \) |
| 29 | \( 1 + 3.97T + 29T^{2} \) |
| 31 | \( 1 + 6.14T + 31T^{2} \) |
| 37 | \( 1 + 8.99T + 37T^{2} \) |
| 41 | \( 1 + 1.01T + 41T^{2} \) |
| 43 | \( 1 - 5.57T + 43T^{2} \) |
| 47 | \( 1 - 3.08T + 47T^{2} \) |
| 53 | \( 1 - 9.11T + 53T^{2} \) |
| 59 | \( 1 + 1.93T + 59T^{2} \) |
| 61 | \( 1 + 0.0788T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 + 3.41T + 73T^{2} \) |
| 79 | \( 1 - 4.66T + 79T^{2} \) |
| 83 | \( 1 + 0.535T + 83T^{2} \) |
| 89 | \( 1 - 0.626T + 89T^{2} \) |
| 97 | \( 1 + 0.616T + 97T^{2} \) |
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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
−10.30443568541006611605833223350, −9.162790287665994712529184460673, −8.388178585660965855963120659894, −7.36484338308494893289986077505, −6.75031280469994919133043036440, −5.31007977731491231373690870311, −4.99446303678337175469996409529, −4.17860568271955509075066930277, −2.22674035838056613497884799487, −0.25159990013742396285648068960,
0.25159990013742396285648068960, 2.22674035838056613497884799487, 4.17860568271955509075066930277, 4.99446303678337175469996409529, 5.31007977731491231373690870311, 6.75031280469994919133043036440, 7.36484338308494893289986077505, 8.388178585660965855963120659894, 9.162790287665994712529184460673, 10.30443568541006611605833223350
