| L(s) = 1 | + (17.9 + 17.9i)2-s + (−30.9 − 74.6i)3-s + 135. i·4-s + (723. − 299. i)5-s + (786. − 1.89e3i)6-s + (5.30e3 + 2.19e3i)7-s + (6.77e3 − 6.77e3i)8-s + (9.30e3 − 9.30e3i)9-s + (1.84e4 + 7.63e3i)10-s + (992. − 2.39e3i)11-s + (1.01e4 − 4.19e3i)12-s + 1.08e5i·13-s + (5.59e4 + 1.35e5i)14-s + (−4.47e4 − 4.47e4i)15-s + 3.13e5·16-s + (−4.94e4 − 3.40e5i)17-s + ⋯ |
| L(s) = 1 | + (0.795 + 0.795i)2-s + (−0.220 − 0.532i)3-s + 0.265i·4-s + (0.517 − 0.214i)5-s + (0.247 − 0.598i)6-s + (0.835 + 0.346i)7-s + (0.584 − 0.584i)8-s + (0.472 − 0.472i)9-s + (0.582 + 0.241i)10-s + (0.0204 − 0.0493i)11-s + (0.141 − 0.0584i)12-s + 1.05i·13-s + (0.389 + 0.939i)14-s + (−0.228 − 0.228i)15-s + 1.19·16-s + (−0.143 − 0.989i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0680i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0680i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.83722 + 0.0966054i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.83722 + 0.0966054i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (4.94e4 + 3.40e5i)T \) |
| good | 2 | \( 1 + (-17.9 - 17.9i)T + 512iT^{2} \) |
| 3 | \( 1 + (30.9 + 74.6i)T + (-1.39e4 + 1.39e4i)T^{2} \) |
| 5 | \( 1 + (-723. + 299. i)T + (1.38e6 - 1.38e6i)T^{2} \) |
| 7 | \( 1 + (-5.30e3 - 2.19e3i)T + (2.85e7 + 2.85e7i)T^{2} \) |
| 11 | \( 1 + (-992. + 2.39e3i)T + (-1.66e9 - 1.66e9i)T^{2} \) |
| 13 | \( 1 - 1.08e5iT - 1.06e10T^{2} \) |
| 19 | \( 1 + (6.76e4 + 6.76e4i)T + 3.22e11iT^{2} \) |
| 23 | \( 1 + (-3.55e4 + 8.58e4i)T + (-1.27e12 - 1.27e12i)T^{2} \) |
| 29 | \( 1 + (2.69e6 - 1.11e6i)T + (1.02e13 - 1.02e13i)T^{2} \) |
| 31 | \( 1 + (8.34e5 + 2.01e6i)T + (-1.86e13 + 1.86e13i)T^{2} \) |
| 37 | \( 1 + (-4.20e6 - 1.01e7i)T + (-9.18e13 + 9.18e13i)T^{2} \) |
| 41 | \( 1 + (-1.02e7 - 4.25e6i)T + (2.31e14 + 2.31e14i)T^{2} \) |
| 43 | \( 1 + (8.53e6 - 8.53e6i)T - 5.02e14iT^{2} \) |
| 47 | \( 1 - 5.88e7iT - 1.11e15T^{2} \) |
| 53 | \( 1 + (-1.31e7 - 1.31e7i)T + 3.29e15iT^{2} \) |
| 59 | \( 1 + (6.73e7 - 6.73e7i)T - 8.66e15iT^{2} \) |
| 61 | \( 1 + (-4.19e7 - 1.73e7i)T + (8.26e15 + 8.26e15i)T^{2} \) |
| 67 | \( 1 + 2.09e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + (-7.85e7 - 1.89e8i)T + (-3.24e16 + 3.24e16i)T^{2} \) |
| 73 | \( 1 + (-1.07e8 + 4.45e7i)T + (4.16e16 - 4.16e16i)T^{2} \) |
| 79 | \( 1 + (-1.49e8 + 3.60e8i)T + (-8.47e16 - 8.47e16i)T^{2} \) |
| 83 | \( 1 + (1.98e8 + 1.98e8i)T + 1.86e17iT^{2} \) |
| 89 | \( 1 + 9.72e8iT - 3.50e17T^{2} \) |
| 97 | \( 1 + (1.18e9 - 4.89e8i)T + (5.37e17 - 5.37e17i)T^{2} \) |
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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
−16.57516923450071437649387193884, −15.22223041120684711245074136635, −14.09874533340132785033413967007, −13.03633953270016581295919687666, −11.55486285864011005435748519808, −9.453775395305938749724147961621, −7.34668127960670120606981047834, −6.04012322343667191240459516006, −4.59101573670021824694023644131, −1.48587779972182454842062164463,
1.94018530672893529688702594336, 3.96316564895821773821668319507, 5.35493903296117498169543823717, 7.906426631340025753147950063333, 10.26933887918283275700563912628, 11.06572429995718164928656603366, 12.68425536976166688700660436597, 13.79771207184641081375907336064, 15.09540577991184725574399726079, 16.81025607323387193553319916896
