| L(s) = 1 | − 8·2-s + 27·3-s + 64·4-s + 438.·5-s − 216·6-s + 343·7-s − 512·8-s + 729·9-s − 3.51e3·10-s − 613.·11-s + 1.72e3·12-s − 2.19e3·13-s − 2.74e3·14-s + 1.18e4·15-s + 4.09e3·16-s + 1.66e4·17-s − 5.83e3·18-s + 1.69e3·19-s + 2.80e4·20-s + 9.26e3·21-s + 4.90e3·22-s − 5.99e4·23-s − 1.38e4·24-s + 1.14e5·25-s + 1.75e4·26-s + 1.96e4·27-s + 2.19e4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.57·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s − 1.11·10-s − 0.138·11-s + 0.288·12-s − 0.277·13-s − 0.267·14-s + 0.906·15-s + 0.250·16-s + 0.822·17-s − 0.235·18-s + 0.0566·19-s + 0.785·20-s + 0.218·21-s + 0.0982·22-s − 1.02·23-s − 0.204·24-s + 1.46·25-s + 0.196·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(3.200103618\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.200103618\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 8T \) |
| 3 | \( 1 - 27T \) |
| 7 | \( 1 - 343T \) |
| 13 | \( 1 + 2.19e3T \) |
| good | 5 | \( 1 - 438.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 613.T + 1.94e7T^{2} \) |
| 17 | \( 1 - 1.66e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.69e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 5.99e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.34e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.46e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.91e3T + 9.49e10T^{2} \) |
| 41 | \( 1 - 4.87e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 8.54e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.10e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 8.24e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 4.45e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 4.81e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 4.20e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.08e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 4.93e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 2.23e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 8.38e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 8.66e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 2.82e6T + 8.07e13T^{2} \) |
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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
−9.486800037540797470315664720266, −9.089035173131829443701098637050, −7.926014118245334399648991315124, −7.21991506349497427456177234953, −5.97020498929535327611344368317, −5.38518008005279235215839234906, −3.83368520651523635282542928311, −2.44134082996618490209067342643, −1.92837205587409255836531146490, −0.849260917156261232667499882814,
0.849260917156261232667499882814, 1.92837205587409255836531146490, 2.44134082996618490209067342643, 3.83368520651523635282542928311, 5.38518008005279235215839234906, 5.97020498929535327611344368317, 7.21991506349497427456177234953, 7.926014118245334399648991315124, 9.089035173131829443701098637050, 9.486800037540797470315664720266
