L(s) = 1 | − 9.57i·3-s + 39.7·5-s + 24.5·7-s + 637.·9-s + 2.03e3·11-s + 2.26e3i·13-s − 380. i·15-s + 3.67e3·17-s + (−5.98e3 + 3.35e3i)19-s − 235. i·21-s + 2.20e3·23-s − 1.40e4·25-s − 1.30e4i·27-s + 2.60e4i·29-s − 5.90e3i·31-s + ⋯ |
L(s) = 1 | − 0.354i·3-s + 0.318·5-s + 0.0716·7-s + 0.874·9-s + 1.53·11-s + 1.02i·13-s − 0.112i·15-s + 0.748·17-s + (−0.872 + 0.489i)19-s − 0.0254i·21-s + 0.180·23-s − 0.898·25-s − 0.664i·27-s + 1.06i·29-s − 0.198i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 - 0.489i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.872 - 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(2.711372986\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.711372986\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (5.98e3 - 3.35e3i)T \) |
good | 3 | \( 1 + 9.57iT - 729T^{2} \) |
| 5 | \( 1 - 39.7T + 1.56e4T^{2} \) |
| 7 | \( 1 - 24.5T + 1.17e5T^{2} \) |
| 11 | \( 1 - 2.03e3T + 1.77e6T^{2} \) |
| 13 | \( 1 - 2.26e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 3.67e3T + 2.41e7T^{2} \) |
| 23 | \( 1 - 2.20e3T + 1.48e8T^{2} \) |
| 29 | \( 1 - 2.60e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 5.90e3iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 1.30e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 - 5.14e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 9.72e3T + 6.32e9T^{2} \) |
| 47 | \( 1 - 1.37e5T + 1.07e10T^{2} \) |
| 53 | \( 1 - 2.74e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 2.31e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 1.22e5T + 5.15e10T^{2} \) |
| 67 | \( 1 + 1.71e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 - 3.78e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 7.65e4T + 1.51e11T^{2} \) |
| 79 | \( 1 + 8.95e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 - 8.96e5T + 3.26e11T^{2} \) |
| 89 | \( 1 - 1.17e6iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 8.02e5iT - 8.32e11T^{2} \) |
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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
−10.75848073840355282495295737364, −9.685166916182621016688208418745, −9.041141866795126242961958622634, −7.79329659570140846026599245998, −6.77083966160895056205510632005, −6.11073426296108408615387006399, −4.57105209942304389772813129833, −3.67868309392216801005961109850, −1.91164577101684717939591557533, −1.18524490735842097846111743304,
0.73038635478468163582540356593, 1.93890415012724710248391117233, 3.52438875853824568037745124846, 4.41300065991162816187653985054, 5.67056774058047976213927566299, 6.66447997987343889898964000459, 7.71804967438006137349997708471, 8.881376166247052401490622855639, 9.752593572406651477295519607447, 10.43346579566218306451243607676