L(s) = 1 | − 30.7i·2-s + (79.3 − 16.0i)3-s − 686.·4-s − 657. i·5-s + (−493. − 2.43e3i)6-s − 1.25e3·7-s + 1.32e4i·8-s + (6.04e3 − 2.55e3i)9-s − 2.01e4·10-s − 4.41e3i·11-s + (−5.45e4 + 1.10e4i)12-s − 4.00e4·13-s + 3.85e4i·14-s + (−1.05e4 − 5.21e4i)15-s + 2.30e5·16-s + 4.88e4i·17-s + ⋯ |
L(s) = 1 | − 1.91i·2-s + (0.980 − 0.198i)3-s − 2.68·4-s − 1.05i·5-s + (−0.380 − 1.88i)6-s − 0.522·7-s + 3.23i·8-s + (0.921 − 0.388i)9-s − 2.01·10-s − 0.301i·11-s + (−2.63 + 0.532i)12-s − 1.40·13-s + 1.00i·14-s + (−0.208 − 1.03i)15-s + 3.51·16-s + 0.584i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(0.884884 + 1.08198i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.884884 + 1.08198i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-79.3 + 16.0i)T \) |
| 11 | \( 1 + 4.41e3iT \) |
good | 2 | \( 1 + 30.7iT - 256T^{2} \) |
| 5 | \( 1 + 657. iT - 3.90e5T^{2} \) |
| 7 | \( 1 + 1.25e3T + 5.76e6T^{2} \) |
| 13 | \( 1 + 4.00e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 4.88e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 1.43e5T + 1.69e10T^{2} \) |
| 23 | \( 1 + 1.27e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 1.23e6iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 1.23e6T + 8.52e11T^{2} \) |
| 37 | \( 1 - 4.93e5T + 3.51e12T^{2} \) |
| 41 | \( 1 - 2.58e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 4.87e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + 3.99e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 7.34e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + 1.78e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 8.49e6T + 1.91e14T^{2} \) |
| 67 | \( 1 - 2.49e6T + 4.06e14T^{2} \) |
| 71 | \( 1 + 4.22e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 9.54e6T + 8.06e14T^{2} \) |
| 79 | \( 1 - 5.39e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + 7.19e6iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 4.49e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 1.04e8T + 7.83e15T^{2} \) |
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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
−13.43515843281697280069279565586, −12.76162298911999811662657767672, −11.83182600830777185422306802760, −9.984602058272973769422535301596, −9.310340226090364596969448349683, −8.118489083477856709644599800532, −4.82038975758198179619748576439, −3.40861267687434282103067991078, −2.01420857269345646771542517224, −0.52990227710366662750441096945,
3.29218434473882363670482875058, 5.07731957072148760857017207991, 6.99895092468996385116136420093, 7.46031794826789668534891949692, 9.135967967357233821208105843915, 10.00444548912668239914372132529, 12.83448663658821626738173907852, 14.15476491993459032324528130971, 14.62323887861419764716863011406, 15.59652508950247214658174097576