| L(s) = 1 | + (−15.6 + 9.01i)2-s + (98.3 − 170. i)4-s − 339. i·5-s + (−437. − 252. i)7-s + 1.23e3i·8-s + (3.06e3 + 5.30e3i)10-s + (−3.64e3 + 2.10e3i)11-s + (−560. + 7.90e3i)13-s + 9.09e3·14-s + (1.42e3 + 2.47e3i)16-s + (−1.37e4 + 2.37e4i)17-s + (4.16e4 + 2.40e4i)19-s + (−5.78e4 − 3.34e4i)20-s + (3.79e4 − 6.56e4i)22-s + (−4.07e4 − 7.05e4i)23-s + ⋯ |
| L(s) = 1 | + (−1.37 + 0.796i)2-s + (0.768 − 1.33i)4-s − 1.21i·5-s + (−0.481 − 0.278i)7-s + 0.855i·8-s + (0.967 + 1.67i)10-s + (−0.825 + 0.476i)11-s + (−0.0707 + 0.997i)13-s + 0.886·14-s + (0.0871 + 0.150i)16-s + (−0.678 + 1.17i)17-s + (1.39 + 0.804i)19-s + (−1.61 − 0.933i)20-s + (0.759 − 1.31i)22-s + (−0.697 − 1.20i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0834i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0834i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.6316606308\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6316606308\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 + (560. - 7.90e3i)T \) |
| good | 2 | \( 1 + (15.6 - 9.01i)T + (64 - 110. i)T^{2} \) |
| 5 | \( 1 + 339. iT - 7.81e4T^{2} \) |
| 7 | \( 1 + (437. + 252. i)T + (4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (3.64e3 - 2.10e3i)T + (9.74e6 - 1.68e7i)T^{2} \) |
| 17 | \( 1 + (1.37e4 - 2.37e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-4.16e4 - 2.40e4i)T + (4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (4.07e4 + 7.05e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (1.72e4 + 2.98e4i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 - 1.20e5iT - 2.75e10T^{2} \) |
| 37 | \( 1 + (-6.06e4 + 3.50e4i)T + (4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + (-5.35e5 + 3.09e5i)T + (9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + (-1.67e5 + 2.89e5i)T + (-1.35e11 - 2.35e11i)T^{2} \) |
| 47 | \( 1 - 1.58e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 - 5.71e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + (-5.55e5 - 3.20e5i)T + (1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-5.20e4 + 9.02e4i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-3.90e5 + 2.25e5i)T + (3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + (-2.65e6 - 1.53e6i)T + (4.54e12 + 7.87e12i)T^{2} \) |
| 73 | \( 1 + 4.62e6iT - 1.10e13T^{2} \) |
| 79 | \( 1 + 7.21e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 5.74e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 + (8.66e6 - 5.00e6i)T + (2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + (-6.35e6 - 3.67e6i)T + (4.03e13 + 6.99e13i)T^{2} \) |
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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
−12.21708074784649634042343113374, −10.58853023437957467481926854433, −9.722894483767350271740329045606, −8.836153051389354049840006955327, −8.017930573620459413064765037365, −6.92659767877588279848864909969, −5.71294867687482550334650801456, −4.22711151791320840192662622461, −1.78061793043090803957188649033, −0.49497515173724444002169839427,
0.67631216745971236512412120469, 2.60887168505860000178164398746, 3.07667271794469609931565041971, 5.56738322530418561578391259914, 7.16400509239311249511519174061, 7.922700432310474985361502295615, 9.356570005444650309476529693378, 9.990087320545196624117998000900, 11.08034452030822807414417420853, 11.52587193100654817171237164150
