| L(s) = 1 | + (−22.5 − 22.5i)2-s + (147. − 60.9i)3-s + 507. i·4-s + (534. + 1.29e3i)5-s + (−4.70e3 − 1.94e3i)6-s + (−3.81e3 + 9.21e3i)7-s + (−95.0 + 95.0i)8-s + (4.03e3 − 4.03e3i)9-s + (1.70e4 − 4.12e4i)10-s + (4.67e4 + 1.93e4i)11-s + (3.09e4 + 7.47e4i)12-s + 7.60e4i·13-s + (2.94e5 − 1.21e5i)14-s + (1.57e5 + 1.57e5i)15-s + 2.64e5·16-s + (−3.24e5 − 1.16e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.997 − 0.997i)2-s + (1.04 − 0.434i)3-s + 0.991i·4-s + (0.382 + 0.923i)5-s + (−1.48 − 0.613i)6-s + (−0.601 + 1.45i)7-s + (−0.00820 + 0.00820i)8-s + (0.204 − 0.204i)9-s + (0.540 − 1.30i)10-s + (0.962 + 0.398i)11-s + (0.431 + 1.04i)12-s + 0.738i·13-s + (2.04 − 0.848i)14-s + (0.803 + 0.803i)15-s + 1.00·16-s + (−0.941 − 0.337i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.384i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.923 - 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.17036 + 0.233734i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.17036 + 0.233734i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (3.24e5 + 1.16e5i)T \) |
| good | 2 | \( 1 + (22.5 + 22.5i)T + 512iT^{2} \) |
| 3 | \( 1 + (-147. + 60.9i)T + (1.39e4 - 1.39e4i)T^{2} \) |
| 5 | \( 1 + (-534. - 1.29e3i)T + (-1.38e6 + 1.38e6i)T^{2} \) |
| 7 | \( 1 + (3.81e3 - 9.21e3i)T + (-2.85e7 - 2.85e7i)T^{2} \) |
| 11 | \( 1 + (-4.67e4 - 1.93e4i)T + (1.66e9 + 1.66e9i)T^{2} \) |
| 13 | \( 1 - 7.60e4iT - 1.06e10T^{2} \) |
| 19 | \( 1 + (4.60e5 + 4.60e5i)T + 3.22e11iT^{2} \) |
| 23 | \( 1 + (-5.47e5 - 2.26e5i)T + (1.27e12 + 1.27e12i)T^{2} \) |
| 29 | \( 1 + (-1.35e6 - 3.26e6i)T + (-1.02e13 + 1.02e13i)T^{2} \) |
| 31 | \( 1 + (-3.15e6 + 1.30e6i)T + (1.86e13 - 1.86e13i)T^{2} \) |
| 37 | \( 1 + (5.75e6 - 2.38e6i)T + (9.18e13 - 9.18e13i)T^{2} \) |
| 41 | \( 1 + (6.43e3 - 1.55e4i)T + (-2.31e14 - 2.31e14i)T^{2} \) |
| 43 | \( 1 + (-2.30e7 + 2.30e7i)T - 5.02e14iT^{2} \) |
| 47 | \( 1 - 5.02e7iT - 1.11e15T^{2} \) |
| 53 | \( 1 + (-1.10e7 - 1.10e7i)T + 3.29e15iT^{2} \) |
| 59 | \( 1 + (1.14e8 - 1.14e8i)T - 8.66e15iT^{2} \) |
| 61 | \( 1 + (-5.96e7 + 1.44e8i)T + (-8.26e15 - 8.26e15i)T^{2} \) |
| 67 | \( 1 - 2.74e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + (-2.51e8 + 1.04e8i)T + (3.24e16 - 3.24e16i)T^{2} \) |
| 73 | \( 1 + (-3.28e6 - 7.92e6i)T + (-4.16e16 + 4.16e16i)T^{2} \) |
| 79 | \( 1 + (2.65e8 + 1.09e8i)T + (8.47e16 + 8.47e16i)T^{2} \) |
| 83 | \( 1 + (-2.62e8 - 2.62e8i)T + 1.86e17iT^{2} \) |
| 89 | \( 1 + 4.29e8iT - 3.50e17T^{2} \) |
| 97 | \( 1 + (-3.90e8 - 9.41e8i)T + (-5.37e17 + 5.37e17i)T^{2} \) |
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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
−17.46551765027565243173555345046, −15.25564923175420198210862739172, −14.13576617516728873750276831695, −12.44599039297222901048498467983, −11.08833253643242115001963699944, −9.349024473491033308641252812860, −8.793830819861337590281705632520, −6.64718640812676655357113829117, −2.82287972206439092731286809097, −2.06525011618868205233201030145,
0.73068119632081491872388729626, 3.84266139549741805275426819344, 6.48544520730006677035168454795, 8.172990628032272817579037470805, 9.106721229228365995813774340734, 10.19471004327263997691071851366, 13.00103592845824748928120094270, 14.28399548455155657945216540379, 15.61239330106851607124464843163, 16.83575301821667813347788529139
