| L(s) = 1 | + 7.11·2-s + 27·3-s − 77.3·4-s − 244.·5-s + 192.·6-s + 549.·7-s − 1.46e3·8-s + 729·9-s − 1.73e3·10-s + 6.49e3·11-s − 2.08e3·12-s + 6.32e3·13-s + 3.90e3·14-s − 6.58e3·15-s − 491.·16-s + 3.56e4·17-s + 5.18e3·18-s + 1.06e4·19-s + 1.88e4·20-s + 1.48e4·21-s + 4.62e4·22-s + 1.21e4·23-s − 3.94e4·24-s − 1.85e4·25-s + 4.50e4·26-s + 1.96e4·27-s − 4.25e4·28-s + ⋯ |
| L(s) = 1 | + 0.628·2-s + 0.577·3-s − 0.604·4-s − 0.872·5-s + 0.363·6-s + 0.605·7-s − 1.00·8-s + 0.333·9-s − 0.548·10-s + 1.47·11-s − 0.349·12-s + 0.798·13-s + 0.380·14-s − 0.504·15-s − 0.0299·16-s + 1.75·17-s + 0.209·18-s + 0.356·19-s + 0.527·20-s + 0.349·21-s + 0.925·22-s + 0.208·23-s − 0.582·24-s − 0.237·25-s + 0.502·26-s + 0.192·27-s − 0.365·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.723257950\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.723257950\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 27T \) |
| 23 | \( 1 - 1.21e4T \) |
| good | 2 | \( 1 - 7.11T + 128T^{2} \) |
| 5 | \( 1 + 244.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 549.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 6.49e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 6.32e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 3.56e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.06e4T + 8.93e8T^{2} \) |
| 29 | \( 1 + 2.53e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.38e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 8.68e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.99e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 2.83e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 3.12e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.01e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 6.77e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.34e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.15e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.36e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.79e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 6.51e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 4.01e5T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.16e7T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.24e7T + 8.07e13T^{2} \) |
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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
−13.46598854070929863127201374708, −12.21365563298716698185096047764, −11.44362471596090100597824220236, −9.636389520946336125321117846808, −8.606450475506536754833236919997, −7.54869709159367535652340487964, −5.75485019671453323538425390603, −4.17275803961446509530753376951, −3.47266567310524062709461960052, −1.09972947380201448561963309220,
1.09972947380201448561963309220, 3.47266567310524062709461960052, 4.17275803961446509530753376951, 5.75485019671453323538425390603, 7.54869709159367535652340487964, 8.606450475506536754833236919997, 9.636389520946336125321117846808, 11.44362471596090100597824220236, 12.21365563298716698185096047764, 13.46598854070929863127201374708
