| L(s) = 1 | − 6.65·2-s − 27·3-s − 83.7·4-s + 355.·5-s + 179.·6-s − 1.54e3·7-s + 1.40e3·8-s + 729·9-s − 2.36e3·10-s + 7.28e3·11-s + 2.26e3·12-s + 7.54e3·13-s + 1.02e4·14-s − 9.58e3·15-s + 1.35e3·16-s + 2.93e4·17-s − 4.84e3·18-s − 5.07e3·19-s − 2.97e4·20-s + 4.17e4·21-s − 4.84e4·22-s + 6.50e4·23-s − 3.80e4·24-s + 4.79e4·25-s − 5.01e4·26-s − 1.96e4·27-s + 1.29e5·28-s + ⋯ |
| L(s) = 1 | − 0.587·2-s − 0.577·3-s − 0.654·4-s + 1.27·5-s + 0.339·6-s − 1.70·7-s + 0.972·8-s + 0.333·9-s − 0.746·10-s + 1.64·11-s + 0.377·12-s + 0.951·13-s + 1.00·14-s − 0.733·15-s + 0.0828·16-s + 1.45·17-s − 0.195·18-s − 0.169·19-s − 0.831·20-s + 0.983·21-s − 0.969·22-s + 1.11·23-s − 0.561·24-s + 0.614·25-s − 0.559·26-s − 0.192·27-s + 1.11·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.730602776\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.730602776\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 27T \) |
| 157 | \( 1 + 3.86e6T \) |
| good | 2 | \( 1 + 6.65T + 128T^{2} \) |
| 5 | \( 1 - 355.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.54e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 7.28e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 7.54e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.93e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 5.07e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 6.50e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 2.08e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.04e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.87e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 1.44e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 5.27e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.09e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 4.14e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.10e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 3.33e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.93e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.78e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.95e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.43e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 6.93e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.76e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.19e6T + 8.07e13T^{2} \) |
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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
−9.873041081108578337481761238366, −9.270396901320723654371916965528, −8.382532768244418305860541530628, −6.72151672505001024786137265069, −6.35800636229612239639654014982, −5.41462886038819156730666282175, −4.09471612987728874356445373715, −3.06134774869288128933433552815, −1.27198247330932576455809915451, −0.814115326587775886956021874113,
0.814115326587775886956021874113, 1.27198247330932576455809915451, 3.06134774869288128933433552815, 4.09471612987728874356445373715, 5.41462886038819156730666282175, 6.35800636229612239639654014982, 6.72151672505001024786137265069, 8.382532768244418305860541530628, 9.270396901320723654371916965528, 9.873041081108578337481761238366
