L(s) = 1 | − 16.1·2-s + 27·3-s + 132.·4-s + 118.·5-s − 436.·6-s + 738.·7-s − 80.6·8-s + 729·9-s − 1.91e3·10-s + 4.27e3·11-s + 3.59e3·12-s − 8.30·13-s − 1.19e4·14-s + 3.19e3·15-s − 1.57e4·16-s − 9.64e3·17-s − 1.17e4·18-s − 1.44e4·19-s + 1.57e4·20-s + 1.99e4·21-s − 6.91e4·22-s + 1.21e4·23-s − 2.17e3·24-s − 6.41e4·25-s + 134.·26-s + 1.96e4·27-s + 9.82e4·28-s + ⋯ |
L(s) = 1 | − 1.42·2-s + 0.577·3-s + 1.03·4-s + 0.423·5-s − 0.824·6-s + 0.813·7-s − 0.0557·8-s + 0.333·9-s − 0.604·10-s + 0.969·11-s + 0.599·12-s − 0.00104·13-s − 1.16·14-s + 0.244·15-s − 0.959·16-s − 0.475·17-s − 0.475·18-s − 0.483·19-s + 0.439·20-s + 0.469·21-s − 1.38·22-s + 0.208·23-s − 0.0321·24-s − 0.820·25-s + 0.00149·26-s + 0.192·27-s + 0.845·28-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
\(L(4)\) |
\(\approx\) |
\(1.413369640\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.413369640\) |
\(L(\frac{9}{2})\) |
|
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 | \( 1 + 16.1T + 128T^{2} \) |
| 5 | \( 1 - 118.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 738.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 4.27e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 8.30T + 6.27e7T^{2} \) |
| 17 | \( 1 + 9.64e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.44e4T + 8.93e8T^{2} \) |
| 29 | \( 1 - 1.98e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 8.05e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.16e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.34e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.62e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 8.78e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 9.71e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 9.16e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 4.44e4T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.44e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.27e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.56e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.09e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 3.98e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 7.23e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 8.46e5T + 8.07e13T^{2} \) |
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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
−13.48291062543730029803601725326, −11.82015057703015761619427725334, −10.68675308360815145206860088849, −9.605975531846220808362365433485, −8.733363503651834486595350141329, −7.83904094196315606154975668989, −6.51631880248209192975429408235, −4.41803420991700869600457114297, −2.22168486101543150311364625944, −1.04312081132085226704607552122,
1.04312081132085226704607552122, 2.22168486101543150311364625944, 4.41803420991700869600457114297, 6.51631880248209192975429408235, 7.83904094196315606154975668989, 8.733363503651834486595350141329, 9.605975531846220808362365433485, 10.68675308360815145206860088849, 11.82015057703015761619427725334, 13.48291062543730029803601725326