| L(s) = 1 | + 4.44e4·2-s − 4.78e6·3-s + 1.44e9·4-s + 4.26e9·5-s − 2.12e11·6-s + 6.78e11·7-s + 4.01e13·8-s + 2.28e13·9-s + 1.89e14·10-s + 3.33e13·11-s − 6.89e15·12-s + 2.47e16·13-s + 3.01e16·14-s − 2.04e16·15-s + 1.01e18·16-s + 3.50e17·17-s + 1.01e18·18-s − 4.86e18·19-s + 6.14e18·20-s − 3.24e18·21-s + 1.48e18·22-s − 6.35e19·23-s − 1.92e20·24-s − 1.68e20·25-s + 1.10e21·26-s − 1.09e20·27-s + 9.77e20·28-s + ⋯ |
| L(s) = 1 | + 1.91·2-s − 0.577·3-s + 2.68·4-s + 0.312·5-s − 1.10·6-s + 0.377·7-s + 3.23·8-s + 0.333·9-s + 0.600·10-s + 0.0264·11-s − 1.54·12-s + 1.74·13-s + 0.725·14-s − 0.180·15-s + 3.51·16-s + 0.504·17-s + 0.639·18-s − 1.39·19-s + 0.839·20-s − 0.218·21-s + 0.0508·22-s − 1.14·23-s − 1.86·24-s − 0.902·25-s + 3.34·26-s − 0.192·27-s + 1.01·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(15)\) |
\(\approx\) |
\(8.806383893\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.806383893\) |
| \(L(\frac{31}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 4.78e6T \) |
| 7 | \( 1 - 6.78e11T \) |
| good | 2 | \( 1 - 4.44e4T + 5.36e8T^{2} \) |
| 5 | \( 1 - 4.26e9T + 1.86e20T^{2} \) |
| 11 | \( 1 - 3.33e13T + 1.58e30T^{2} \) |
| 13 | \( 1 - 2.47e16T + 2.01e32T^{2} \) |
| 17 | \( 1 - 3.50e17T + 4.81e35T^{2} \) |
| 19 | \( 1 + 4.86e18T + 1.21e37T^{2} \) |
| 23 | \( 1 + 6.35e19T + 3.09e39T^{2} \) |
| 29 | \( 1 - 1.29e21T + 2.56e42T^{2} \) |
| 31 | \( 1 - 6.01e21T + 1.77e43T^{2} \) |
| 37 | \( 1 - 1.93e22T + 3.00e45T^{2} \) |
| 41 | \( 1 - 1.89e23T + 5.89e46T^{2} \) |
| 43 | \( 1 - 6.08e23T + 2.34e47T^{2} \) |
| 47 | \( 1 + 5.68e23T + 3.09e48T^{2} \) |
| 53 | \( 1 + 1.42e25T + 1.00e50T^{2} \) |
| 59 | \( 1 - 7.70e25T + 2.26e51T^{2} \) |
| 61 | \( 1 + 4.19e25T + 5.95e51T^{2} \) |
| 67 | \( 1 + 4.17e26T + 9.04e52T^{2} \) |
| 71 | \( 1 - 6.10e25T + 4.85e53T^{2} \) |
| 73 | \( 1 - 5.13e26T + 1.08e54T^{2} \) |
| 79 | \( 1 - 3.72e27T + 1.07e55T^{2} \) |
| 83 | \( 1 - 3.79e27T + 4.50e55T^{2} \) |
| 89 | \( 1 - 9.29e27T + 3.40e56T^{2} \) |
| 97 | \( 1 + 7.85e28T + 4.13e57T^{2} \) |
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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
−12.32829358684807003931717383951, −11.33308219738814753739095151903, −10.42244772798844619334074556856, −8.006756090599704897162551739604, −6.31579964077282729691535524869, −5.95501847245031568571425602477, −4.57661723860691897384767003076, −3.75474658681915674625014110675, −2.29514670413861352688568348528, −1.19724159682022932055312109468,
1.19724159682022932055312109468, 2.29514670413861352688568348528, 3.75474658681915674625014110675, 4.57661723860691897384767003076, 5.95501847245031568571425602477, 6.31579964077282729691535524869, 8.006756090599704897162551739604, 10.42244772798844619334074556856, 11.33308219738814753739095151903, 12.32829358684807003931717383951
