| L(s) = 1 | + 10.2·2-s + 27·3-s − 23.7·4-s − 397.·5-s + 275.·6-s − 505.·7-s − 1.54e3·8-s + 729·9-s − 4.05e3·10-s − 5.49e3·11-s − 642.·12-s + 5.26e3·13-s − 5.16e3·14-s − 1.07e4·15-s − 1.27e4·16-s + 3.36e4·17-s + 7.44e3·18-s + 4.18e4·19-s + 9.46e3·20-s − 1.36e4·21-s − 5.60e4·22-s − 1.47e4·23-s − 4.18e4·24-s + 8.00e4·25-s + 5.37e4·26-s + 1.96e4·27-s + 1.20e4·28-s + ⋯ |
| L(s) = 1 | + 0.902·2-s + 0.577·3-s − 0.185·4-s − 1.42·5-s + 0.520·6-s − 0.557·7-s − 1.07·8-s + 0.333·9-s − 1.28·10-s − 1.24·11-s − 0.107·12-s + 0.664·13-s − 0.502·14-s − 0.821·15-s − 0.779·16-s + 1.66·17-s + 0.300·18-s + 1.39·19-s + 0.264·20-s − 0.321·21-s − 1.12·22-s − 0.252·23-s − 0.617·24-s + 1.02·25-s + 0.599·26-s + 0.192·27-s + 0.103·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.067525245\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.067525245\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 27T \) |
| 59 | \( 1 - 2.05e5T \) |
| good | 2 | \( 1 - 10.2T + 128T^{2} \) |
| 5 | \( 1 + 397.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 505.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 5.49e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 5.26e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 3.36e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.18e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 1.47e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.92e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.22e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.19e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 2.10e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 1.97e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.24e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 3.46e5T + 1.17e12T^{2} \) |
| 61 | \( 1 + 2.78e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 4.61e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.49e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.38e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 7.13e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 2.03e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 5.52e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.50e7T + 8.07e13T^{2} \) |
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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
−11.82604023333759167043368010968, −10.44752245594387757352348214344, −9.300610833843144949506683226228, −8.112712198793032521076140097378, −7.47509579047455931165323794589, −5.83673531977203186419235700212, −4.71860929748580653526884039620, −3.48902821247304872028428202814, −3.08696547030317116808625043204, −0.66240305464672685674841881425,
0.66240305464672685674841881425, 3.08696547030317116808625043204, 3.48902821247304872028428202814, 4.71860929748580653526884039620, 5.83673531977203186419235700212, 7.47509579047455931165323794589, 8.112712198793032521076140097378, 9.300610833843144949506683226228, 10.44752245594387757352348214344, 11.82604023333759167043368010968
