| L(s) = 1 | − 21.6·2-s + 27·3-s + 342.·4-s − 369.·5-s − 585.·6-s − 1.00e3·7-s − 4.64e3·8-s + 729·9-s + 8.01e3·10-s + 9.23e3·12-s + 1.27e4·13-s + 2.16e4·14-s − 9.98e3·15-s + 5.68e4·16-s + 3.58e3·17-s − 1.58e4·18-s + 4.99e4·19-s − 1.26e5·20-s − 2.70e4·21-s + 9.27e3·23-s − 1.25e5·24-s + 5.86e4·25-s − 2.76e5·26-s + 1.96e4·27-s − 3.42e5·28-s − 1.07e5·29-s + 2.16e5·30-s + ⋯ |
| L(s) = 1 | − 1.91·2-s + 0.577·3-s + 2.67·4-s − 1.32·5-s − 1.10·6-s − 1.10·7-s − 3.20·8-s + 0.333·9-s + 2.53·10-s + 1.54·12-s + 1.61·13-s + 2.11·14-s − 0.764·15-s + 3.46·16-s + 0.176·17-s − 0.638·18-s + 1.67·19-s − 3.53·20-s − 0.636·21-s + 0.158·23-s − 1.85·24-s + 0.751·25-s − 3.09·26-s + 0.192·27-s − 2.94·28-s − 0.815·29-s + 1.46·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.5959556608\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5959556608\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 27T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 21.6T + 128T^{2} \) |
| 5 | \( 1 + 369.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.00e3T + 8.23e5T^{2} \) |
| 13 | \( 1 - 1.27e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 3.58e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.99e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 9.27e3T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.07e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 9.43e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.77e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 3.85e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 6.69e4T + 2.71e11T^{2} \) |
| 47 | \( 1 - 4.07e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 4.18e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 8.17e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.47e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 8.86e4T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.96e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 6.16e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 3.36e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 2.92e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 5.54e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 4.30e6T + 8.07e13T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.975935884806170745519577364026, −9.150485981989343001850254964605, −8.518791377490272735408239397306, −7.61856462708852891232245882662, −7.06816308707770873715311340217, −5.94615580900540261699651067377, −3.59183622193585715975593938944, −3.12304030584119867231314258078, −1.49066482897220250708199718492, −0.49368913590802682179465343028,
0.49368913590802682179465343028, 1.49066482897220250708199718492, 3.12304030584119867231314258078, 3.59183622193585715975593938944, 5.94615580900540261699651067377, 7.06816308707770873715311340217, 7.61856462708852891232245882662, 8.518791377490272735408239397306, 9.150485981989343001850254964605, 9.975935884806170745519577364026
