| L(s) = 1 | + (−9.79 + 5.65i)2-s + (−57.0 + 32.9i)3-s + (63.9 − 110. i)4-s + (−412. − 714. i)5-s + (372. − 645. i)6-s + 2.84e3·7-s + 1.44e3i·8-s + (−1.10e3 + 1.91e3i)9-s + (8.08e3 + 4.66e3i)10-s + 1.88e4·11-s + 8.43e3i·12-s + (−3.66e4 − 2.11e4i)13-s + (−2.78e4 + 1.60e4i)14-s + (4.70e4 + 2.71e4i)15-s + (−8.19e3 − 1.41e4i)16-s + (−3.85e4 − 6.67e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.612 + 0.353i)2-s + (−0.704 + 0.406i)3-s + (0.249 − 0.433i)4-s + (−0.659 − 1.14i)5-s + (0.287 − 0.498i)6-s + 1.18·7-s + 0.353i·8-s + (−0.168 + 0.292i)9-s + (0.808 + 0.466i)10-s + 1.29·11-s + 0.406i·12-s + (−1.28 − 0.740i)13-s + (−0.725 + 0.418i)14-s + (0.930 + 0.537i)15-s + (−0.125 − 0.216i)16-s + (−0.461 − 0.799i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.788 - 0.615i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.788 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.126558 + 0.367836i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.126558 + 0.367836i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (9.79 - 5.65i)T \) |
| 19 | \( 1 + (5.37e3 - 1.30e5i)T \) |
| good | 3 | \( 1 + (57.0 - 32.9i)T + (3.28e3 - 5.68e3i)T^{2} \) |
| 5 | \( 1 + (412. + 714. i)T + (-1.95e5 + 3.38e5i)T^{2} \) |
| 7 | \( 1 - 2.84e3T + 5.76e6T^{2} \) |
| 11 | \( 1 - 1.88e4T + 2.14e8T^{2} \) |
| 13 | \( 1 + (3.66e4 + 2.11e4i)T + (4.07e8 + 7.06e8i)T^{2} \) |
| 17 | \( 1 + (3.85e4 + 6.67e4i)T + (-3.48e9 + 6.04e9i)T^{2} \) |
| 23 | \( 1 + (2.51e5 - 4.34e5i)T + (-3.91e10 - 6.78e10i)T^{2} \) |
| 29 | \( 1 + (3.16e5 + 1.82e5i)T + (2.50e11 + 4.33e11i)T^{2} \) |
| 31 | \( 1 - 1.48e6iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 1.34e6iT - 3.51e12T^{2} \) |
| 41 | \( 1 + (2.99e6 - 1.72e6i)T + (3.99e12 - 6.91e12i)T^{2} \) |
| 43 | \( 1 + (-2.31e6 - 4.00e6i)T + (-5.84e12 + 1.01e13i)T^{2} \) |
| 47 | \( 1 + (-3.04e6 + 5.27e6i)T + (-1.19e13 - 2.06e13i)T^{2} \) |
| 53 | \( 1 + (-5.85e6 - 3.38e6i)T + (3.11e13 + 5.39e13i)T^{2} \) |
| 59 | \( 1 + (1.85e6 - 1.07e6i)T + (7.34e13 - 1.27e14i)T^{2} \) |
| 61 | \( 1 + (7.79e6 - 1.35e7i)T + (-9.58e13 - 1.66e14i)T^{2} \) |
| 67 | \( 1 + (1.34e7 + 7.77e6i)T + (2.03e14 + 3.51e14i)T^{2} \) |
| 71 | \( 1 + (1.52e7 - 8.82e6i)T + (3.22e14 - 5.59e14i)T^{2} \) |
| 73 | \( 1 + (-1.46e7 - 2.54e7i)T + (-4.03e14 + 6.98e14i)T^{2} \) |
| 79 | \( 1 + (5.16e7 - 2.98e7i)T + (7.58e14 - 1.31e15i)T^{2} \) |
| 83 | \( 1 - 3.53e7T + 2.25e15T^{2} \) |
| 89 | \( 1 + (5.54e7 + 3.20e7i)T + (1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 + (1.55e7 - 8.99e6i)T + (3.91e15 - 6.78e15i)T^{2} \) |
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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
−15.30196168760057224552836712745, −14.09574708075069352528911925877, −12.04525788146166243826606544682, −11.51386945016981497995179851731, −9.971500509534854554182500731754, −8.578565066391810825440675860187, −7.56397468251102662293875215482, −5.46460812970045289153368700434, −4.52678672200920399264055855438, −1.35964601795929665082968607764,
0.22094860711757046611757383146, 2.09771429744904712475739804195, 4.16938822036623700846181469001, 6.48881789867667254439932369654, 7.42902187084258802137433673606, 8.994764854812530773541121491556, 10.74804675950719315826699478885, 11.54401855131164923199177199390, 12.19119957249400439501053160163, 14.39015563484819132445947481901
