| L(s) = 1 | + (7.18 − 8.73i)2-s + (25.5 − 61.5i)3-s + (−24.7 − 125. i)4-s + (−98.9 + 41.0i)5-s + (−354. − 665. i)6-s + (436. − 436. i)7-s + (−1.27e3 − 686. i)8-s + (−1.59e3 − 1.59e3i)9-s + (−353. + 1.15e3i)10-s + (2.05e3 + 4.95e3i)11-s + (−8.36e3 − 1.68e3i)12-s + (2.91e3 + 1.20e3i)13-s + (−676. − 6.94e3i)14-s + 7.13e3i·15-s + (−1.51e4 + 6.20e3i)16-s − 2.31e4i·17-s + ⋯ |
| L(s) = 1 | + (0.635 − 0.772i)2-s + (0.545 − 1.31i)3-s + (−0.193 − 0.981i)4-s + (−0.354 + 0.146i)5-s + (−0.670 − 1.25i)6-s + (0.480 − 0.480i)7-s + (−0.880 − 0.474i)8-s + (−0.728 − 0.728i)9-s + (−0.111 + 0.366i)10-s + (0.464 + 1.12i)11-s + (−1.39 − 0.280i)12-s + (0.367 + 0.152i)13-s + (−0.0659 − 0.676i)14-s + 0.546i·15-s + (−0.925 + 0.378i)16-s − 1.14i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.958 + 0.285i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.958 + 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.375073 - 2.57459i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.375073 - 2.57459i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-7.18 + 8.73i)T \) |
| good | 3 | \( 1 + (-25.5 + 61.5i)T + (-1.54e3 - 1.54e3i)T^{2} \) |
| 5 | \( 1 + (98.9 - 41.0i)T + (5.52e4 - 5.52e4i)T^{2} \) |
| 7 | \( 1 + (-436. + 436. i)T - 8.23e5iT^{2} \) |
| 11 | \( 1 + (-2.05e3 - 4.95e3i)T + (-1.37e7 + 1.37e7i)T^{2} \) |
| 13 | \( 1 + (-2.91e3 - 1.20e3i)T + (4.43e7 + 4.43e7i)T^{2} \) |
| 17 | \( 1 + 2.31e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 + (3.60e3 + 1.49e3i)T + (6.32e8 + 6.32e8i)T^{2} \) |
| 23 | \( 1 + (1.24e4 + 1.24e4i)T + 3.40e9iT^{2} \) |
| 29 | \( 1 + (-9.41e4 + 2.27e5i)T + (-1.21e10 - 1.21e10i)T^{2} \) |
| 31 | \( 1 - 1.86e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + (-6.23e4 + 2.58e4i)T + (6.71e10 - 6.71e10i)T^{2} \) |
| 41 | \( 1 + (-5.53e5 - 5.53e5i)T + 1.94e11iT^{2} \) |
| 43 | \( 1 + (-2.28e4 - 5.52e4i)T + (-1.92e11 + 1.92e11i)T^{2} \) |
| 47 | \( 1 + 7.61e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 + (8.89e4 + 2.14e5i)T + (-8.30e11 + 8.30e11i)T^{2} \) |
| 59 | \( 1 + (2.62e6 - 1.08e6i)T + (1.75e12 - 1.75e12i)T^{2} \) |
| 61 | \( 1 + (-6.36e5 + 1.53e6i)T + (-2.22e12 - 2.22e12i)T^{2} \) |
| 67 | \( 1 + (-1.01e6 + 2.45e6i)T + (-4.28e12 - 4.28e12i)T^{2} \) |
| 71 | \( 1 + (3.42e6 - 3.42e6i)T - 9.09e12iT^{2} \) |
| 73 | \( 1 + (-3.45e6 - 3.45e6i)T + 1.10e13iT^{2} \) |
| 79 | \( 1 - 7.08e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 + (-5.01e6 - 2.07e6i)T + (1.91e13 + 1.91e13i)T^{2} \) |
| 89 | \( 1 + (1.92e6 - 1.92e6i)T - 4.42e13iT^{2} \) |
| 97 | \( 1 + 3.08e6T + 8.07e13T^{2} \) |
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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
−14.19807587538335593751878686080, −13.49892669860638826748284355854, −12.29188946485127928786532858312, −11.40684456697473625038419617930, −9.666898268903603018828644638580, −7.85425745035393903488097247654, −6.59193648425001119978416572183, −4.37492608260256462490898527807, −2.43824157866675576956588838568, −1.04553048550292160569416432393,
3.32323840549052734806055838239, 4.45776644504705580273464555787, 5.97411668140934430965626360370, 8.212284573805008279382049731872, 8.945424217314961831026643296515, 10.79822874195966369253226679463, 12.22588646947803483869177640084, 13.89914310642287285073483716921, 14.76649710772411857811151995471, 15.71506916005341188585553809934
