| L(s) = 1 | + (5.25 − 5.25i)2-s + (−18.5 − 7.70i)3-s − 23.1i·4-s + (32.1 − 77.6i)5-s + (−138. + 57.2i)6-s + (39.9 + 96.4i)7-s + (46.3 + 46.3i)8-s + (114. + 114. i)9-s + (−238. − 576. i)10-s + (403. − 167. i)11-s + (−178. + 431. i)12-s + 285. i·13-s + (716. + 296. i)14-s + (−1.19e3 + 1.19e3i)15-s + 1.22e3·16-s + (−414. − 1.11e3i)17-s + ⋯ |
| L(s) = 1 | + (0.928 − 0.928i)2-s + (−1.19 − 0.494i)3-s − 0.724i·4-s + (0.575 − 1.38i)5-s + (−1.56 + 0.648i)6-s + (0.308 + 0.743i)7-s + (0.256 + 0.256i)8-s + (0.472 + 0.472i)9-s + (−0.755 − 1.82i)10-s + (1.00 − 0.416i)11-s + (−0.357 + 0.864i)12-s + 0.468i·13-s + (0.976 + 0.404i)14-s + (−1.37 + 1.37i)15-s + 1.19·16-s + (−0.347 − 0.937i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.374 + 0.927i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.924580 - 1.37023i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.924580 - 1.37023i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (414. + 1.11e3i)T \) |
| good | 2 | \( 1 + (-5.25 + 5.25i)T - 32iT^{2} \) |
| 3 | \( 1 + (18.5 + 7.70i)T + (171. + 171. i)T^{2} \) |
| 5 | \( 1 + (-32.1 + 77.6i)T + (-2.20e3 - 2.20e3i)T^{2} \) |
| 7 | \( 1 + (-39.9 - 96.4i)T + (-1.18e4 + 1.18e4i)T^{2} \) |
| 11 | \( 1 + (-403. + 167. i)T + (1.13e5 - 1.13e5i)T^{2} \) |
| 13 | \( 1 - 285. iT - 3.71e5T^{2} \) |
| 19 | \( 1 + (2.06e3 - 2.06e3i)T - 2.47e6iT^{2} \) |
| 23 | \( 1 + (-1.94e3 + 806. i)T + (4.55e6 - 4.55e6i)T^{2} \) |
| 29 | \( 1 + (1.72e3 - 4.15e3i)T + (-1.45e7 - 1.45e7i)T^{2} \) |
| 31 | \( 1 + (-683. - 283. i)T + (2.02e7 + 2.02e7i)T^{2} \) |
| 37 | \( 1 + (-2.78e3 - 1.15e3i)T + (4.90e7 + 4.90e7i)T^{2} \) |
| 41 | \( 1 + (3.69e3 + 8.91e3i)T + (-8.19e7 + 8.19e7i)T^{2} \) |
| 43 | \( 1 + (-8.35e3 - 8.35e3i)T + 1.47e8iT^{2} \) |
| 47 | \( 1 - 8.98e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 + (2.63e4 - 2.63e4i)T - 4.18e8iT^{2} \) |
| 59 | \( 1 + (5.82e3 + 5.82e3i)T + 7.14e8iT^{2} \) |
| 61 | \( 1 + (1.43e4 + 3.47e4i)T + (-5.97e8 + 5.97e8i)T^{2} \) |
| 67 | \( 1 - 1.95e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (5.37e4 + 2.22e4i)T + (1.27e9 + 1.27e9i)T^{2} \) |
| 73 | \( 1 + (-4.30e3 + 1.03e4i)T + (-1.46e9 - 1.46e9i)T^{2} \) |
| 79 | \( 1 + (7.01e4 - 2.90e4i)T + (2.17e9 - 2.17e9i)T^{2} \) |
| 83 | \( 1 + (1.67e4 - 1.67e4i)T - 3.93e9iT^{2} \) |
| 89 | \( 1 + 3.52e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (1.03e4 - 2.50e4i)T + (-6.07e9 - 6.07e9i)T^{2} \) |
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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
−17.23329291610604464441485335920, −16.60002996882233015215211954123, −14.24043030370710452725422534174, −12.76266790593704668511125713728, −12.19218921380455603415356064931, −11.14878423399011450739212115774, −8.920784235469100454746738264657, −5.96407325100865394043438799366, −4.73548316029771320260706221972, −1.48964988438567783759435118928,
4.35441447029966040978700537000, 6.07498565218630061111508736858, 6.95528652395701712791971207665, 10.27336243654991013267440734326, 11.16464582397197850170680379847, 13.23563174997457843304682764654, 14.57572778420797857991532285523, 15.34383100189593217011985410358, 17.08847410140210824422467869165, 17.46569762994110266984358659619
