| L(s) = 1 | + 0.737·2-s − 2.81·3-s − 1.45·4-s − 5-s − 2.07·6-s − 1.03·7-s − 2.54·8-s + 4.90·9-s − 0.737·10-s + 4.09·12-s − 3.44·13-s − 0.760·14-s + 2.81·15-s + 1.03·16-s − 2.39·17-s + 3.61·18-s + 7.66·19-s + 1.45·20-s + 2.89·21-s + 2.45·23-s + 7.16·24-s + 25-s − 2.54·26-s − 5.35·27-s + 1.50·28-s + 5.95·29-s + 2.07·30-s + ⋯ |
| L(s) = 1 | + 0.521·2-s − 1.62·3-s − 0.727·4-s − 0.447·5-s − 0.846·6-s − 0.389·7-s − 0.901·8-s + 1.63·9-s − 0.233·10-s + 1.18·12-s − 0.956·13-s − 0.203·14-s + 0.725·15-s + 0.257·16-s − 0.581·17-s + 0.852·18-s + 1.75·19-s + 0.325·20-s + 0.632·21-s + 0.512·23-s + 1.46·24-s + 0.200·25-s − 0.498·26-s − 1.03·27-s + 0.283·28-s + 1.10·29-s + 0.378·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5944325774\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5944325774\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.737T + 2T^{2} \) |
| 3 | \( 1 + 2.81T + 3T^{2} \) |
| 7 | \( 1 + 1.03T + 7T^{2} \) |
| 13 | \( 1 + 3.44T + 13T^{2} \) |
| 17 | \( 1 + 2.39T + 17T^{2} \) |
| 19 | \( 1 - 7.66T + 19T^{2} \) |
| 23 | \( 1 - 2.45T + 23T^{2} \) |
| 29 | \( 1 - 5.95T + 29T^{2} \) |
| 31 | \( 1 + 3.68T + 31T^{2} \) |
| 37 | \( 1 - 5.95T + 37T^{2} \) |
| 41 | \( 1 + 3.93T + 41T^{2} \) |
| 43 | \( 1 - 7.64T + 43T^{2} \) |
| 47 | \( 1 - 5.84T + 47T^{2} \) |
| 53 | \( 1 + 11.8T + 53T^{2} \) |
| 59 | \( 1 - 2.94T + 59T^{2} \) |
| 61 | \( 1 + 2.48T + 61T^{2} \) |
| 67 | \( 1 + 6.14T + 67T^{2} \) |
| 71 | \( 1 - 2.02T + 71T^{2} \) |
| 73 | \( 1 - 0.825T + 73T^{2} \) |
| 79 | \( 1 - 12.0T + 79T^{2} \) |
| 83 | \( 1 + 1.61T + 83T^{2} \) |
| 89 | \( 1 - 8.16T + 89T^{2} \) |
| 97 | \( 1 - 2.44T + 97T^{2} \) |
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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
−10.85528140569375451734048062061, −9.852783618135527405012184989951, −9.190677256165423477948855746747, −7.76453987913318987175574630660, −6.82163698092168120350873667973, −5.88521181159174531679615875841, −5.04383778001913608768518182240, −4.47081105832619136421584044285, −3.14893485823379414840832848208, −0.66691259212154053818966833566,
0.66691259212154053818966833566, 3.14893485823379414840832848208, 4.47081105832619136421584044285, 5.04383778001913608768518182240, 5.88521181159174531679615875841, 6.82163698092168120350873667973, 7.76453987913318987175574630660, 9.190677256165423477948855746747, 9.852783618135527405012184989951, 10.85528140569375451734048062061
