| L(s) = 1 | − 14.4·2-s − 27·3-s + 81.2·4-s + 346.·5-s + 390.·6-s − 1.59e3·7-s + 676.·8-s + 729·9-s − 5.01e3·10-s − 5.16e3·11-s − 2.19e3·12-s + 1.12e4·13-s + 2.30e4·14-s − 9.35e3·15-s − 2.01e4·16-s − 7.24e3·17-s − 1.05e4·18-s − 5.46e4·19-s + 2.81e4·20-s + 4.30e4·21-s + 7.46e4·22-s − 5.63e3·23-s − 1.82e4·24-s + 4.20e4·25-s − 1.63e5·26-s − 1.96e4·27-s − 1.29e5·28-s + ⋯ |
| L(s) = 1 | − 1.27·2-s − 0.577·3-s + 0.634·4-s + 1.24·5-s + 0.738·6-s − 1.75·7-s + 0.467·8-s + 0.333·9-s − 1.58·10-s − 1.16·11-s − 0.366·12-s + 1.42·13-s + 2.24·14-s − 0.716·15-s − 1.23·16-s − 0.357·17-s − 0.426·18-s − 1.82·19-s + 0.786·20-s + 1.01·21-s + 1.49·22-s − 0.0966·23-s − 0.269·24-s + 0.538·25-s − 1.82·26-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
| \(L(4)\) |
\(\approx\) |
\(0.2552518675\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2552518675\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 + 14.4T + 128T^{2} \) |
| 5 | \( 1 - 346.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.59e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 5.16e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.12e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 7.24e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 5.46e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 5.63e3T + 3.40e9T^{2} \) |
| 29 | \( 1 + 9.58e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.01e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.28e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 7.75e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 1.71e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 8.68e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.01e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.48e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.97e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.08e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.52e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.92e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.24e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 3.44e5T + 2.71e13T^{2} \) |
| 89 | \( 1 + 6.50e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.15e7T + 8.07e13T^{2} \) |
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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
−9.867282797053133899161691878109, −9.144593669765578442804656206446, −8.375299382515209524583547172356, −7.03317467769551870023369859407, −6.26365639218825395402574065132, −5.64910518992804525744599118669, −4.08252884700175203559880925585, −2.59922200375259082624284900647, −1.58986944505283591696372823440, −0.28032854068773968322669848453,
0.28032854068773968322669848453, 1.58986944505283591696372823440, 2.59922200375259082624284900647, 4.08252884700175203559880925585, 5.64910518992804525744599118669, 6.26365639218825395402574065132, 7.03317467769551870023369859407, 8.375299382515209524583547172356, 9.144593669765578442804656206446, 9.867282797053133899161691878109
