L(s) = 1 | + (9.79 − 5.65i)2-s + (63.9 − 110. i)4-s + (−683. − 394. i)5-s + (89.7 + 155. i)7-s − 1.44e3i·8-s − 8.92e3·10-s + (4.61e3 − 2.66e3i)11-s + (−2.51e4 + 4.35e4i)13-s + (1.75e3 + 1.01e3i)14-s + (−8.19e3 − 1.41e4i)16-s + 1.16e5i·17-s + 7.72e3·19-s + (−8.74e4 + 5.04e4i)20-s + (3.01e4 − 5.21e4i)22-s + (−3.58e5 − 2.07e5i)23-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (−1.09 − 0.631i)5-s + (0.0373 + 0.0647i)7-s − 0.353i·8-s − 0.892·10-s + (0.314 − 0.181i)11-s + (−0.879 + 1.52i)13-s + (0.0457 + 0.0264i)14-s + (−0.125 − 0.216i)16-s + 1.39i·17-s + 0.0592·19-s + (−0.546 + 0.315i)20-s + (0.128 − 0.222i)22-s + (−1.28 − 0.740i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.444 - 0.895i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(0.222162 + 0.358065i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.222162 + 0.358065i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-9.79 + 5.65i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (683. + 394. i)T + (1.95e5 + 3.38e5i)T^{2} \) |
| 7 | \( 1 + (-89.7 - 155. i)T + (-2.88e6 + 4.99e6i)T^{2} \) |
| 11 | \( 1 + (-4.61e3 + 2.66e3i)T + (1.07e8 - 1.85e8i)T^{2} \) |
| 13 | \( 1 + (2.51e4 - 4.35e4i)T + (-4.07e8 - 7.06e8i)T^{2} \) |
| 17 | \( 1 - 1.16e5iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 7.72e3T + 1.69e10T^{2} \) |
| 23 | \( 1 + (3.58e5 + 2.07e5i)T + (3.91e10 + 6.78e10i)T^{2} \) |
| 29 | \( 1 + (3.18e5 - 1.83e5i)T + (2.50e11 - 4.33e11i)T^{2} \) |
| 31 | \( 1 + (7.02e5 - 1.21e6i)T + (-4.26e11 - 7.38e11i)T^{2} \) |
| 37 | \( 1 - 1.79e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + (1.27e6 + 7.35e5i)T + (3.99e12 + 6.91e12i)T^{2} \) |
| 43 | \( 1 + (-9.01e5 - 1.56e6i)T + (-5.84e12 + 1.01e13i)T^{2} \) |
| 47 | \( 1 + (3.93e6 - 2.27e6i)T + (1.19e13 - 2.06e13i)T^{2} \) |
| 53 | \( 1 - 8.53e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + (1.46e7 + 8.45e6i)T + (7.34e13 + 1.27e14i)T^{2} \) |
| 61 | \( 1 + (7.89e6 + 1.36e7i)T + (-9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (-1.57e7 + 2.72e7i)T + (-2.03e14 - 3.51e14i)T^{2} \) |
| 71 | \( 1 + 4.80e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 1.30e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + (-1.73e7 - 3.01e7i)T + (-7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 + (-2.32e6 + 1.34e6i)T + (1.12e15 - 1.95e15i)T^{2} \) |
| 89 | \( 1 + 3.46e6iT - 3.93e15T^{2} \) |
| 97 | \( 1 + (-1.54e7 - 2.67e7i)T + (-3.91e15 + 6.78e15i)T^{2} \) |
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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
−14.05679738856145286702793780478, −12.50731264305958403241673528424, −12.01715640666761170797571782029, −10.84096040887647941342448063654, −9.269066972807591501461075940282, −7.966782824740716459030607404208, −6.44966111440585644953492886414, −4.69451969799484402523051889627, −3.78390187354681791564859734198, −1.76711131601169937168228179510,
0.11482713464017094499340796144, 2.79054448507197729591063731189, 4.05496410545442784001119244377, 5.56594740640633293980225485928, 7.26293511442371323741413434880, 7.86366881794643924281158201249, 9.802148758071111891274673907509, 11.29356520931612394126419140949, 12.07528386012163624734462877612, 13.33743175462593683685808346980