| L(s) = 1 | + 2.79·2-s − 1.08·3-s + 5.82·4-s − 1.79·5-s − 3.02·6-s + 10.7·8-s − 1.82·9-s − 5.02·10-s − 11-s − 6.31·12-s − 3.74·13-s + 1.94·15-s + 18.3·16-s + 17-s − 5.11·18-s + 0.314·19-s − 10.4·20-s − 2.79·22-s − 2.51·23-s − 11.5·24-s − 1.76·25-s − 10.4·26-s + 5.22·27-s + 5.02·29-s + 5.44·30-s − 8.42·31-s + 29.7·32-s + ⋯ |
| L(s) = 1 | + 1.97·2-s − 0.625·3-s + 2.91·4-s − 0.803·5-s − 1.23·6-s + 3.78·8-s − 0.609·9-s − 1.59·10-s − 0.301·11-s − 1.82·12-s − 1.03·13-s + 0.502·15-s + 4.57·16-s + 0.242·17-s − 1.20·18-s + 0.0722·19-s − 2.34·20-s − 0.596·22-s − 0.523·23-s − 2.36·24-s − 0.353·25-s − 2.05·26-s + 1.00·27-s + 0.933·29-s + 0.994·30-s − 1.51·31-s + 5.26·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.570550095\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.570550095\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 - 2.79T + 2T^{2} \) |
| 3 | \( 1 + 1.08T + 3T^{2} \) |
| 5 | \( 1 + 1.79T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 13 | \( 1 + 3.74T + 13T^{2} \) |
| 19 | \( 1 - 0.314T + 19T^{2} \) |
| 23 | \( 1 + 2.51T + 23T^{2} \) |
| 29 | \( 1 - 5.02T + 29T^{2} \) |
| 31 | \( 1 + 8.42T + 31T^{2} \) |
| 37 | \( 1 - 8.51T + 37T^{2} \) |
| 41 | \( 1 - 4.16T + 41T^{2} \) |
| 43 | \( 1 + 7.48T + 43T^{2} \) |
| 47 | \( 1 - 4.88T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 - 6.93T + 59T^{2} \) |
| 61 | \( 1 - 13.1T + 61T^{2} \) |
| 67 | \( 1 + 7.07T + 67T^{2} \) |
| 71 | \( 1 + 15.2T + 71T^{2} \) |
| 73 | \( 1 + 0.506T + 73T^{2} \) |
| 79 | \( 1 - 4.05T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 + 3.14T + 89T^{2} \) |
| 97 | \( 1 + 4.84T + 97T^{2} \) |
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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
−12.47931840440334233060891105725, −11.83607275540918176058469082572, −11.26228900430737501721726080393, −10.21522665507085653947612532728, −7.967462393495367360834145349164, −7.04276389536795903900129601864, −5.87186506170867011865989347179, −5.03673871208632593263595656968, −3.94370513385051033933376544764, −2.62374977390360810583160140079,
2.62374977390360810583160140079, 3.94370513385051033933376544764, 5.03673871208632593263595656968, 5.87186506170867011865989347179, 7.04276389536795903900129601864, 7.967462393495367360834145349164, 10.21522665507085653947612532728, 11.26228900430737501721726080393, 11.83607275540918176058469082572, 12.47931840440334233060891105725
