| L(s) = 1 | + (64 − 110. i)2-s + (675 + 1.16e3i)3-s + (−8.19e3 − 1.41e4i)4-s + (−4.05e4 + 7.02e4i)5-s + 1.72e5·6-s − 2.09e6·8-s + (6.26e6 − 1.08e7i)9-s + (5.18e6 + 8.98e6i)10-s + (−3.50e7 − 6.07e7i)11-s + (1.10e7 − 1.91e7i)12-s − 1.51e8·13-s − 1.09e8·15-s + (−1.34e8 + 2.32e8i)16-s + (−1.24e8 − 2.16e8i)17-s + (−8.01e8 − 1.38e9i)18-s + (−3.23e9 + 5.60e9i)19-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (0.178 + 0.308i)3-s + (−0.249 − 0.433i)4-s + (−0.232 + 0.401i)5-s + 0.252·6-s − 0.353·8-s + (0.436 − 0.756i)9-s + (0.164 + 0.284i)10-s + (−0.542 − 0.939i)11-s + (0.0890 − 0.154i)12-s − 0.669·13-s − 0.165·15-s + (−0.125 + 0.216i)16-s + (−0.0738 − 0.127i)17-s + (−0.308 − 0.534i)18-s + (−0.831 + 1.43i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(0.9560215283\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9560215283\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-64 + 110. i)T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + (-675 - 1.16e3i)T + (-7.17e6 + 1.24e7i)T^{2} \) |
| 5 | \( 1 + (4.05e4 - 7.02e4i)T + (-1.52e10 - 2.64e10i)T^{2} \) |
| 11 | \( 1 + (3.50e7 + 6.07e7i)T + (-2.08e15 + 3.61e15i)T^{2} \) |
| 13 | \( 1 + 1.51e8T + 5.11e16T^{2} \) |
| 17 | \( 1 + (1.24e8 + 2.16e8i)T + (-1.43e18 + 2.47e18i)T^{2} \) |
| 19 | \( 1 + (3.23e9 - 5.60e9i)T + (-7.59e18 - 1.31e19i)T^{2} \) |
| 23 | \( 1 + (-1.05e10 + 1.82e10i)T + (-1.33e20 - 2.30e20i)T^{2} \) |
| 29 | \( 1 - 7.79e9T + 8.62e21T^{2} \) |
| 31 | \( 1 + (4.75e10 + 8.23e10i)T + (-1.17e22 + 2.03e22i)T^{2} \) |
| 37 | \( 1 + (-4.35e11 + 7.53e11i)T + (-1.66e23 - 2.88e23i)T^{2} \) |
| 41 | \( 1 + 1.00e12T + 1.55e24T^{2} \) |
| 43 | \( 1 - 1.55e11T + 3.17e24T^{2} \) |
| 47 | \( 1 + (1.27e12 - 2.21e12i)T + (-6.03e24 - 1.04e25i)T^{2} \) |
| 53 | \( 1 + (2.02e12 + 3.50e12i)T + (-3.65e25 + 6.33e25i)T^{2} \) |
| 59 | \( 1 + (6.29e12 + 1.09e13i)T + (-1.82e26 + 3.16e26i)T^{2} \) |
| 61 | \( 1 + (1.99e13 - 3.45e13i)T + (-3.01e26 - 5.21e26i)T^{2} \) |
| 67 | \( 1 + (-2.42e13 - 4.19e13i)T + (-1.23e27 + 2.13e27i)T^{2} \) |
| 71 | \( 1 - 3.76e13T + 5.87e27T^{2} \) |
| 73 | \( 1 + (-7.07e13 - 1.22e14i)T + (-4.45e27 + 7.71e27i)T^{2} \) |
| 79 | \( 1 + (1.23e14 - 2.13e14i)T + (-1.45e28 - 2.52e28i)T^{2} \) |
| 83 | \( 1 + 2.78e12T + 6.11e28T^{2} \) |
| 89 | \( 1 + (2.91e12 - 5.05e12i)T + (-8.70e28 - 1.50e29i)T^{2} \) |
| 97 | \( 1 + 2.78e14T + 6.33e29T^{2} \) |
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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
−11.12973533621763143427124834701, −10.36865412745688384613940773403, −9.343959409982033912574067504623, −8.192069890403047713434594315098, −6.79999973336743260385600699199, −5.63442421330361100394424900495, −4.31009397193365133253779637199, −3.40331714622553046311154377026, −2.42297210940002269414051782037, −0.960193328010726589950142995793,
0.17948585390830573245080548452, 1.75521708108634609960758265808, 2.88713041762529480946845144533, 4.60570863197327328220350968495, 4.98148167779246493289087515245, 6.68853904601569035620109167338, 7.47293665332678465988708653820, 8.390910594947497029384560368608, 9.592925651464611535322842641407, 10.82424672117595284072389978417
