| L(s) = 1 | − 13.0·2-s + 27·3-s + 43.1·4-s + 38.5·5-s − 353.·6-s − 1.33e3·7-s + 1.11e3·8-s + 729·9-s − 504.·10-s + 1.16e3·12-s − 1.30e4·13-s + 1.74e4·14-s + 1.04e3·15-s − 2.00e4·16-s − 1.32e4·17-s − 9.53e3·18-s + 1.58e4·19-s + 1.66e3·20-s − 3.59e4·21-s + 4.98e4·23-s + 2.99e4·24-s − 7.66e4·25-s + 1.70e5·26-s + 1.96e4·27-s − 5.73e4·28-s − 1.34e5·29-s − 1.36e4·30-s + ⋯ |
| L(s) = 1 | − 1.15·2-s + 0.577·3-s + 0.336·4-s + 0.137·5-s − 0.667·6-s − 1.46·7-s + 0.766·8-s + 0.333·9-s − 0.159·10-s + 0.194·12-s − 1.64·13-s + 1.69·14-s + 0.0796·15-s − 1.22·16-s − 0.655·17-s − 0.385·18-s + 0.530·19-s + 0.0464·20-s − 0.846·21-s + 0.854·23-s + 0.442·24-s − 0.980·25-s + 1.89·26-s + 0.192·27-s − 0.493·28-s − 1.02·29-s − 0.0920·30-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
| \(L(4)\) |
\(\approx\) |
\(0.3954490191\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3954490191\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 + 13.0T + 128T^{2} \) |
| 5 | \( 1 - 38.5T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.33e3T + 8.23e5T^{2} \) |
| 13 | \( 1 + 1.30e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.32e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.58e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 4.98e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.34e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.89e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.40e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.99e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 9.17e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.04e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.28e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.94e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 4.90e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.38e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.03e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.42e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 3.47e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 3.35e5T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.00e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.77e6T + 8.07e13T^{2} \) |
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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
−9.787694940006605185456029873008, −9.449104501863357289347816770679, −8.675660323999060561200535514861, −7.34330624461677680687228282818, −7.05778750211663025146939030713, −5.49118573601798714623146464389, −4.12872695157464417723802181919, −2.89594920744874977238118221928, −1.85534004702121657934599071160, −0.32844099540793814658323609878,
0.32844099540793814658323609878, 1.85534004702121657934599071160, 2.89594920744874977238118221928, 4.12872695157464417723802181919, 5.49118573601798714623146464389, 7.05778750211663025146939030713, 7.34330624461677680687228282818, 8.675660323999060561200535514861, 9.449104501863357289347816770679, 9.787694940006605185456029873008
