| L(s) = 1 | − 3.63·2-s − 3·3-s + 5.18·4-s + 21.1·5-s + 10.8·6-s + 7·7-s + 10.2·8-s + 9·9-s − 76.6·10-s + 11·11-s − 15.5·12-s + 87.3·13-s − 25.4·14-s − 63.3·15-s − 78.6·16-s − 28.2·17-s − 32.6·18-s − 97.9·19-s + 109.·20-s − 21·21-s − 39.9·22-s − 112.·23-s − 30.6·24-s + 320.·25-s − 317.·26-s − 27·27-s + 36.2·28-s + ⋯ |
| L(s) = 1 | − 1.28·2-s − 0.577·3-s + 0.647·4-s + 1.88·5-s + 0.741·6-s + 0.377·7-s + 0.452·8-s + 0.333·9-s − 2.42·10-s + 0.301·11-s − 0.373·12-s + 1.86·13-s − 0.485·14-s − 1.09·15-s − 1.22·16-s − 0.403·17-s − 0.427·18-s − 1.18·19-s + 1.22·20-s − 0.218·21-s − 0.387·22-s − 1.01·23-s − 0.261·24-s + 2.56·25-s − 2.39·26-s − 0.192·27-s + 0.244·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.167835244\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.167835244\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 7 | \( 1 - 7T \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 + 3.63T + 8T^{2} \) |
| 5 | \( 1 - 21.1T + 125T^{2} \) |
| 13 | \( 1 - 87.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 28.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 97.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 112.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 14.9T + 2.43e4T^{2} \) |
| 31 | \( 1 - 138.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 206.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 321.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 285.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 303.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 554.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 693.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 156.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 584.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 363.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 747.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 419.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.17e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 397.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.33e3T + 9.12e5T^{2} \) |
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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
−11.12559454992077890728740463258, −10.68135051395737917094302759166, −9.742425860901052985000236764127, −9.001533901847018301292413133202, −8.118103597015872081320266812156, −6.45045136289045120851465561453, −6.03016613445236212322849883334, −4.48850284240629286807991395865, −2.05081071765436100564573873773, −1.09844256276300120927701175377,
1.09844256276300120927701175377, 2.05081071765436100564573873773, 4.48850284240629286807991395865, 6.03016613445236212322849883334, 6.45045136289045120851465561453, 8.118103597015872081320266812156, 9.001533901847018301292413133202, 9.742425860901052985000236764127, 10.68135051395737917094302759166, 11.12559454992077890728740463258
